The $p$-adic local invariant cycle theorem

Let $R$ be a complete discrete valuation ring with fraction field $K$ and finite residue field $k$ of characteristic $p$, and let $X$ be a proper flat scheme over $S=\operatorname{Spec}(R)$. Write $X_s$ for its special fibre and $X_{\bar{\eta}}$ for its geometric generic fibre. Let

$$\operatorname{sp}':H^i_{\operatorname{rig}}(X_s/k)\longrightarrow D_{\operatorname{crys}}\bigl(H^i(X_{\bar{\eta}},\mathbb{Q}_p)\bigr)$$

be the $p$-adic specialization morphism, and let $W_j$ be the Frobenius-stable weight filtrations on the source and target. $p$-adic local invariant cycle theorem. If $X$ is regular, then $\operatorname{sp}'$ is an epimorphism, induces an isomorphism on $W_{i-1}$, and is an isomorphism for $i=0,1$. The conjecture is known for semistable $X$ in the stated low-dimensional and low-degree cases, while the general regular case remains open.