Answer: No! Not really! It is extremely rare, virtually impossible. A decadent pagan, a despairing nihilist, or a violent Marxist is more likely to convert than a true "Catholic" Progressive. Think of the legacy American Progressive Catholic families: Kennedy, Cuomo, Kerry, Biden, Peolosi! Do we hear of family members making a good confession and embarking on a life of chastity, or monastic solitude, or service directly with the poor? Hardly! Of course, we know that all things are possible to God; so we pray for them!
Let's contrast a Progressive and a Catholic and then a Catholic Progressive and a Progressive Catholic!
The Progressive looks for guidance, not to the ancient past, but to recent and especially anticipated future developments in science and technology. The past, as ignorant and deprived, is to be overcome by PROGRESS, which is eventually infallible. Contemporary, post 1968 progressivism, is the merging of five streams: evolutionary Darwinism which posits an inexorable "arc of history" fueled by science, reason, and technology; post-Freudian sexual liberation that locates human flourishing in the release of sterilized sex from marriage-family-children-the moral order, and God's creative and providential purposes; the ever present Marxist dialectic of oppression and liberation; more recently a pantheist infatuation with Mother Nature detached from a Creator-Father; and above all the Nietzchean sovereignty of the isolated Self. These five views, if articulated clearly, become mutually contradictory, but philosophical depth and lucidity are foreign to feeling-based progressivism.
A Catholic looks for guidance to the person-event of Jesus Christ who revealed the Trinity in Israel two millennia ago and remains in living intimacy with the Catholic Church in prayer, teaching, and life. This is an organic, living, interpersonal, dynamic, and dramatic reality with a distinctive, eventful history. But throughout time there abides a stable, permanent reality that expresses itself creatively and surprisingly.
A Catholic Progressive is a progressive with Catholic flavoring. Interiorly, in heart-soul-form-substance, the defining gestalt is that of progress. Any Catholic baggage that does not go with the flow of sexual/political liberation and "science" is displaced: contraception, heteronormativity, trans-phobia, etc. Such a "cafeteria" or "Catholic lite" progressive retains the elements of Catholicism that are compatible with the underlying paradigm: conspicuous marriage in a beautiful Church, burial, support of liberal social justice initiatives, and such.
A Progressive Catholic by contrast is interiorly Catholic: specifically, in a filial posture of trust, gratitude and obedience to the hierarchical Church. The defining center of ones life remains Christ present in his Church. Such a Catholic might very well be progressive on any number of current controversies in culture and politics: the environment, ecumenism, governmental action for the poor, liturgical reform, doctrinal development, gun control, taxes, immigration, foreign policy and health care. Such would not violate core Catholic values, especially involving the protection of powerless human life, sexuality-gender-family, and the moral limits of science. Clear denial of those principles, as in advocacy for legal abortion or "choice" is a step out of the Catholic into the Progressive religion. Underlying it is a disparaging judgment against the Church.
To understand the progressive resistance to Catholicism let us compare it with Islam. Both are Christian heresies which take elements of Divine Revelation and reject others. Mohammed, in Arabia of 600 AD, was familiar with Judaism and Christianity, especially the Arian form which denied the divinity of Christ. Spiritual genius that he was, he concocted a brilliant synthesis of elements from local paganism, Christianity and Judaism. Particularly, he incorporated monotheism and the moral code of the ten commandments. This is part of the inner core of the religion. It derives from Divine Revelation and is therefore good and true. Now the bad news: he rejected monogamy in favor of a misogynistic polygamy. He strongly employed the use of violence and force to spread his new religion. Implicitly this included an anti-intellectualism and a fundamentalist fideism. He denied the divinity, the crucifixion and the salvation by Christ and also the Trinity. His acceptance of monotheism was iconoclastic and anti-Christian as it disparaged as polytheism the doctrine of the Trinity. We see very few converts from Islam into Christianity. There are many reasons for this: in some countries converts are executed; everywhere converts are rejected by the community. But interiorly or spiritually there is an even stronger resistance: the religion has a powerful, deep, true core to the extent that it honor the Creator God and the moral code of the 10 commandments. From this high ground, it despises Christianity as polytheistic and idolatrous.
A similar dynamic is at work in Catholic Progressivism. It does not self-define as a rejection of the Church, but as an enlightened, higher expression of it. It condemns the actual, institutional Church as repressive sexually, as misogynistic, patriarchal, reactionary. And so it is inherently immunized against fundamental Catholic truths: binary gender as an image of God, the fertile intentionality of sexuality, moral order, inherent evils, apostolic authority, ecclesial infallibility, and sacramental efficacy.
St. Charles de Focauld labored heroically in the Sahara for decades and was widely revered as a holy man, but he did not have a single convert. Muslim immunity against the Christian Gospel is deep. And so it is with the Catholic Progressive. Both faiths retain aspects of Revelation; both pervert them into an alternative religion, superior to actual Catholicism. And so like Focauld, we live peacefully, patiently in the dessert of Progressivism, commending our neighbors to God's love, extending always reverence and compassion, leaning ever more deeply into our Lord living in the Church.