The Jesuits want to expand the boundaries of what constitutes a “Catholic school”… but only in their direction. But if the Church stretches the boundaries for one, it will for others, if they are elastic now. There are schools identifying as Catholic that reject Vatican 2, reject racial equality, endorse fundamentalism, promote Bayside apparitions, or snake handling, whatever.
The Jesuits would not support elasticity in other directions, nor would most of us. Furthermore, if you he Church’s definition of Catholic School becomes strictly by self report, the definition of Jesuit School could be a anyone who wants to call themselves such, the same criteria could be used for"Jesuit School".
Any school can identify itself as “Jesuit” based on their interpretation of St Ignatius.