The regular-model specialization conjecture for rigid and crystalline cohomology

Let $X\to S$ be proper, flat, generically smooth and regular. Let $\operatorname{sp}'$ be the specialization map from rigid cohomology to crystalline cohomology, and let $W_{i-1}$ denote the indicated weight filtration. Regular-model specialization conjecture. The map $\operatorname{sp}'$ is surjective; the induced map

$$W_{i-1}H^i_{\operatorname{rig}}(X_s/k)\xrightarrow{\operatorname{sp}'}W_{i-1}D_{\operatorname{cris}}(H^i(X_{\bar\eta},\mathbb Q_p))$$

is an isomorphism; and $\operatorname{sp}'$ is an isomorphism for $i=0,1$. This is described as the $p$-adic analogue of the local theorem on invariant cycles, conditional on regularity.