The following comes from an April 4 Catholic News Agency article by Ann Schneible:
Both the Vatican and the Society of St. Pius X have confirmed that this weekend Pope Francis met with the priestly society’s superior general, Bishop Bernard Fellay, at the Vatican.
An April 4 statement issued by the SSPX, meanwhile, said that Pope Francis received Bishop Fellay at his Domus Sanctae Marthae residence in the Vatican on April 1. The statement said Bishop Fellay was accompanied by Fr. Alain-Marc Nely, second assistant general of the priestly society.
“Pope Francis had wanted a private and informal meeting, without the formality of an official audience. It lasted 40 minutes and took place under a cordial atmosphere,” the SSPX stated. “After the meeting, it was decided that the current exchanges would continue. The canonical status of the Society was not directly addressed, Pope Francis and Bishop Fellay having determined that these exchanges ought to continue without haste.”
The biggest obstacle for the society’s reconciliation has been the statements on religious liberty in Vatican II’s declaration Dignitatis Humanae, which it claims contradicts previous Catholic teaching.
In 2015 the Holy See delegated a cardinal and three bishops to visit the seminaries of the SSPX. They were sent to become better acquainted with the society, and to discuss doctrinal and theological topics in a less formal context.
And Pope Francis announced in a September 2015 letter on the Jubilee Year of Mercy that during the jubilee year the faithful can validly and licitly receive absolution of their sins from priests of the SSPX.
“I trust that in the near future solutions may be found to recover full communion with the priests and superiors of the Fraternity,” he wrote.