Doran–Thompson mirror Clemens–Schmid conjecture

Let $X$ be an $n$-dimensional Calabi–Yau manifold undergoing a Type $N$ degeneration $\mathcal{X}\to\Delta$, where Type $N$ means that the logarithm of monodromy has nilpotency index $N$. Let $Y$ be the mirror Calabi–Yau manifold, and let $\pi\colon Y\to B$ be a fibration whose fibres have codimension $N-1$. Write $F$, $W$, and $P$ for the Hodge, weight, and perverse Leray filtrations, respectively. Doran–Thompson's mirror Clemens–Schmid conjecture. The graded pieces satisfy

$$\dim\bigl(\operatorname{Gr}_F^p\operatorname{Gr}^W_q\operatorname{Gr}^P_lH^k(Y)\bigr)=\dim\bigl(\operatorname{Gr}_F^{n-p}\operatorname{Gr}^W_{n+l-2p}\operatorname{Gr}^P_{n+q-2p}H^{n+k-2p}_{\lim}(X)\bigr).$$

Moreover, for an appropriate embedding of $B$ into projective space, if $U\subset Y$ is the complement of the preimage of a general linear subspace, then

$$\dim\bigl(\operatorname{Gr}_F^p\operatorname{Gr}^W_q\operatorname{Gr}^P_lH^k_c(U)\bigr)=\dim\bigl(\operatorname{Gr}_F^{n-p}\operatorname{Gr}^W_{n+l-2p}\operatorname{Gr}^P_{n+q-2p+1}H^{n+k-2p}(\mathcal{X})\bigr).$$

The conjecture proposes that the Clemens–Schmid sequence for the degeneration and the mirror Clemens–Schmid sequence for the fibration are mirror to one another. The paper states that the relationship is exhibited explicitly for all Type II and many Type III degenerations of K3 surfaces, but the general claim is not resolved in the supplied text.