Hodge-theoretic adjacency conjecture for k-Du Bois degenerations

Let $\mathfrak{X}\to\Delta$ be a flat, projective degeneration with reduced special fiber $X_0$, and let $\pi\colon\mathfrak{X}^*\to\Delta^*$ be smooth, where $\mathfrak{X}^*:=\mathfrak{X}\setminus X_0$. Assume that $\mathfrak{X}$ embeds as an open analytic subset of a projective algebraic variety equipped with a flat projective morphism to a smooth curve. For a projective variety $X_0$ with a hyper-resolution $\epsilon\colon\widetilde{X}_{0,\bullet}\to X_0$, set

$$\widetilde{\Omega}^p_{X_0}:=\operatorname{Gr}_F^p(R\epsilon_*\Omega_{\widetilde{X}_{0,\bullet}}^{\bullet})[p].$$

The variety $X_0$ has $k$-Du Bois singularities if the natural map $\Omega^p_{X_0}\to\widetilde{\Omega}^p_{X_0}$ is a quasi-isomorphism for $0\leq p\leq k$. Hodge-theoretic adjacency conjecture. If $X_0$ has $k$-Du Bois singularities, then

$$(H^r(X_0))^{p,q}\cong (H^r_{\lim}(X_t))^{p,q}$$

for $0\leq p\leq k$ and all $q,r$, with trivial $T^{\mathrm{ss}}$-action on the right-hand side. This predicts that the indicated Hodge components of the special fiber agree with those of the limiting cohomology, while the semisimple part of monodromy acts trivially on the latter. The surrounding discussion presents this as a proposed generalization of inversion of adjunction and of results for smooth total spaces; the parser supplies no evidence that it has been proved or disproved.