Affine compactification conjecture for degenerating enhanced projective families

Let $T$ be an affine variety carrying a universal full family of enhanced projective varieties. Let $f:Y\to S$ be a family of projective varieties with fixed prescribed topological data, let $\Delta\subset S$ be its discriminant variety, and allow the family to be neither enhanced nor smooth over $\Delta$. Over the complement of the discriminant, the morphism

$$Y\backslash f^{-1}(\Delta)\to S\backslash\Delta$$

is the underlying morphism of an enhanced family and therefore determines a map $S\backslash\Delta\to T$. The affine compactification conjecture asserts that there is an affine variety $\tilde T\supset T$ of the same dimension as $T$ such that this map extends to

$$S\to\tilde T.$$

The conjecture concerns the existence of $\tilde T$ with this extension property for every such family. It would provide a parameter space accommodating degenerations of projective varieties and extend the quasi-modular form framework beyond smooth enhanced families.