Bishop Bernard Fellay, Superior General of the Society of St. Pius X met with the Holy Father on April 1, then with Archbishop Guido Pozzo on April 2. A few days after, La Croix published the following comments of Abp. Pozzo.

About the Meeting between the Pope and Bishop Fellay
For Abp. Pozzo, secretary for the Ecclesia Dei Commission that takes care of the relations with Tradition within the Congregation for the Doctrine of the Faith, the meeting on April 1 between Pope Francis and Bp. Fellay happened

in the context of the SSPX’s path towards full reconciliation, which will come with a canonical recognition of the Institution.”

At this point, it is important above all to help create an atmosphere of trust (…) in order to rise above the tension and distrust that can be understandable after so many years of distance and fracture,” continued Abp. Pozzo, who declared his wish to "dissipate" this tension and distrust in order to once again find “the reasons for unity and the promotion of the integrity of the Catholic Faith and the Tradition of the Church.”
What is Demanded from a Catholic?
It is important to recall the three essential points that make a person Catholic:

adherence to the profession of the Faith, the bond of the sacraments, and hierarchical communion with the Pope.”
Adherence to the Second Vatican Council?
Abp. Pozzo explained to La Croix:

As far as the Second Vatican Council is concerned, the ground covered in the meetings over the past few years has led to an important clarification: Vatican II can be adequately understood only in the context of the full Tradition of the Church and her constant Magisterium.”
For example, among the Declarations of the Second Vatican Council, Nostra Aetate —which opened the door to the new ecuмenism with the non-Christian religions— is now considered by the secretary of the Ecclesia Dei Comission as

directives for pastoral action, directions, and suggestions or exhortations of a practical pastoral nature.

"The difficulties raised by the SSPX concerning the Church-State relationship and religious freedom, the practice of ecuмenism and dialogue with non-Christian religions, certain aspects of the liturgical reform and its concrete application, remain subject to discussion and clarification but do not constitute an obstacle to a canonical and juridical recognition of the SSPX.

"The docuмents of the Second Vatican Council must be received with the required degree of adherence.

"The acceptance of the texts on relations with other religions is not a prerequisite for the canonical recognition of the Lefebvrist society, and certain questions can remain 'subject to discussion and clarification'."
A Canonical Recognition?
Anticipating already a canonical recognition —for which he has been working since 1987— the prelate announces that the Second Vatican Declarations

will, even after the canonical recognition, remain subject to discussion and deeper study, in order to obtain greater precision and avoid the misunderstandings or ambivalences that we know to have spread throughout today’s ecclesial world.”
According to Abp. Pozzo, the SSPX is requested

to accept that the Magisterium of the Church is the only one entrusted with guarding, defending and interpreting the deposit of the Faith.

"I think that this clarification can be a fixed point for the SSPX.”
On April 11, 2016, Bp. Fellay commented during a sermon in front of 4,000 pilgrims in Le Puy:

I actually think this change is a profound one, a very important one; it has come about because of the Church’s dire situation. One might say it is in part the result of the chaos that is taking hold of the Church. there is such confusion, there are such attacks against faith, against morality in every way that finally, it is as if the Congregation of the Faith decided: We have no right to treat in such a way those people who only say and teach what the Church has always taught....we have no right to consider what they do as a very serious sin, while all around them there are so many, even prelates and cardinals - we would almost say all the way up to the Pope - who not only talk nonsense, but utter heresies that are an open path to sin.

"Yet there are some men in the Church who are reacting, thinking, who are saying: things should not be like this. And it is in the midst of this disorder, amid tears that comes this whisper: 'No, we cannot force you to accept the Council.' They perhaps will not say it so clearly, but they did indeed say it to us after all. Of course, we take this very cautiously, we ask God to enlighten us so that we see what it all means, if it's really true, of whether tomorrow it will all start again in another direction. Nevertheless, my dear brothers, all this shows us something: that fidelity to all that the Church has always taught really does pay. We must simply remain firm. These modern people cannot deny it. The reality is obvious: we are Catholics and we want to remain so."

http://sspx.org/en/news-events/news/new-prerequisites-sspx-abp-pozzo-15229

"For Abp. Pozzo, secretary for the Ecclesia Dei Commission that takes care of the relations with Tradition within the Congregation for the Doctrine of the Faith, the meeting on April 1 between Pope Francis and Bp. Fellay happened in the context of the SSPX’s path towards full reconciliation, which will come with a canonical recognition of the Institution."