In the same spirit and with the same commitment to encouraging the resolution of all fractures and divisions in the Church and to healing a wound in the ecclesial fabric that was more and more painfully felt, I wished to remit the excommunication of the four Bishops illicitly ordained by Archbishop Lefebvre. With this decision I intended to remove an impediment that might have jeopardized the opening of a door to dialogue and thereby to invite the Bishops and the "Society of St. Pius X" to rediscover the path to full communion with the Church. As I explained in my Letter to the Catholic Bishops of last 10 March, the remission of the excommunication was a measure taken in the context of ecclesiastical discipline to free the individuals from the burden of conscience constituted by the most serious of ecclesiastical penalties. However, the doctrinal questions obviously remain and until they are clarified the Society has no canonical status in the Church and its ministers cannot legitimately exercise any ministry. (emphasis added)
The SSPX bishops were excommunicated and their illicit consecrations defined as schismatic in the motu proprio Ecclesia Dei issued July 2, 1988:
In itself, this act was one of disobedience to the Roman Pontiff in a very grave matter and of supreme importance for the unity of the Church, such as is the ordination of bishops whereby the apostolic succession is sacramentally perpetuated. Hence such disobedience — which implies in practice the rejection of the Roman primacy — constitutes a schismatic act. (emphasis added)
The excommunications have been lifted, but the canonical status of the SSPX remains unchanged.
There are two important points here:
1. The SSPX is not in full communion with the Church and is invited by the Church to rediscover this path.
2. The SSPX has no canonical status in the Church and its ministers cannot legitimately exercise any ministry.
This has never been formally changed by the Holy See, in spite of Pope Francis' recent indults allowing the Society to witness marriages or hear confessions. The benefit was provided explicitly for the good of the faithful in the Jubilee Year of Mercy, and not for the benefit of the SSPX.
Many Catholics are unaware that marriages witnessed in the SSPX before the indult resulted in thousands of invalid marriages. Those who wished to return to full communion with the Church were required to have their marriages convalidated.
For the benefit of such Catholics, the pope offered the indult for SSPX clergy to validly preside over marriages — in order to prevent the grave problem of Catholics entering into invalid (therefore sinful) unions. Thus the 2009 motu proprio declaring the SSPX to have "no canonical status" remains in force and has never changed.
While SSPX loyalists love to cite Pope Francis' indult as "proof" the SSPX cannot be in schism, recall that the Orthodox Church also has faculties to witness marriages and hear confessions, yet no Catholic would claim the Orthodox Church is not in schism. This is simply not an argument that can be made.
Some mistakenly argue that Pope Benedict's 2009 lifting of the excommunications on four SSPX bishops ended the schism. Not true. Benedict's lifting of the excommunications did nothing to change the SSPX's objective state of schism, any more than Pope Paul VI's 1965 lifting of excommunications imposed on Orthodox bishops affected the Orthodox Church's state of schism.
In the letter accompanying Traditionis Custodes, Pope Francis also refers to the SSPX as in schism:
The faculty — granted by the indult of the Congregation for Divine Worship in 1984 and confirmed by St. John Paul II in the Motu Proprio Ecclesia Dei in 1988 — was above all motivated by the desire to foster the healing of the schism with the movement of Abp. Lefebvre. (emphasis added)
[T]he fact of the matter is that the Priestly Society of St. Pius X is in schism since the late Abp. Marcel Lefebvre ordained four bishops without the mandate of the Roman Pontiff. And so it is not legitimate to attend Mass or to receive the sacraments in a church that's under the direction of the Priestly Society of St. Pius X.
Cardinal Gerhard Müller, former prefect of the Congregation for the Doctrine of the Faith, has also said the SSPX is in schism in comments made in 2013 to the Italian press:
The canonical excommunication of the bishops for their illegal ordinations was revoked, but a de facto sacramental excommunication remains for their schism; they put themselves out of communion with the Church. After that we are not closing the door and never will, but we are inviting them to be reconciled. But they too must change their attitude, accept the conditions of the Catholic Church, and the Supreme Pontiff as the definitive criterion for membership.
Communion with the Chair of Peter is a constitutive part of Roman Catholicism — something the SSPX accepts in principle but rejects in practice. As Pope Benedict said, the SSPX has "no canonical status," "no legitimate ministry," is "invited to rediscover the path to full communion," has no faculties from any bishop, is not part of the divinely ordained hierarchical structure of the Church and all its priests are suspended a divinis.
We are well aware of ongoing dialogue between the SSPX and Rome. It is to be fervently hoped that this dialogue results in a return of the SSPX to full communion with the Church, granting its bishops and clergy canonical status and the authority to exercise ministry. Reconciliation of the SSPX with the Church would be a great blessing for the Church but most especially for the SSPX.
The SSPX has consistently refused the most generous and magnanimous invitations from the Church to reconcile and place itself under the authority of the Vicar of Christ, rejecting his dogmatically defined "full power of shepherding, ruling and governing the universal Church," a power "ordinary and immediate over all the churches and over each and every member of the faithful" ("The Pope Exercises Supreme Jurisdiction" [Original Italian here]).
Perhaps the SSPX fears it will be crushed by an unsympathetic hierarchy and pope. Perhaps it is justified in such fearful apprehensions. But there is precedent in the history of the Church — in the lives of the saints, in the life of Our Lord Himself — for risking or allowing oneself to be crushed by obedience. Until such time as the SSPX chooses to reconcile with the Church, Church Militant must support the judgment of the Holy Father and the Church concerning its status.
"My Journey out of the Lefebvre Schism" (external website)