The p-adic local invariant cycle conjecture

Let $R$ be a henselian discrete valuation ring with finite residue field, let $R^{sh}$ be a strict henselization, and let $K$ and $K^{sh}$ be the fraction fields of $R$ and $R^{sh}$. Let $I=\operatorname{Gal}(\overline K/K^{sh})$. For a proper flat morphism $\pi:{\mathcal X}\longrightarrow S=\operatorname{Spec}R$ with ${\mathcal X}$ regular, let $X$ be its generic fiber, and define

$$H^i_L({\mathcal X}_{R^{sh}},\mathbb{Q}_\ell(m)):=\varprojlim_n \mathbb{H}^i_{\operatorname{et}}({\mathcal X}_{R^{sh}},\mathbb{Z}(m)_{\operatorname{et}}\otimes^{\mathbb L}\mathbb{Z}/\ell^n)\otimes_{\mathbb Z_\ell}\mathbb Q_\ell.$$

The p-adic local invariant cycle conjecture. If the residue field of $R$ is finite, there is a surjective natural map

$$H^i_L({\mathcal X}_{R^{sh}},\mathbb{Q}_\ell(m))\longrightarrow H^i_{\operatorname{et}}(X_{\overline K},\mathbb{Q}_\ell(m))^I.$$

This extends the local invariant cycle principle to the motivic or étale Tate-twist setting; the source explains that, under purity of the weight filtration, related surjectivity follows in known cases, but the stated general conjecture remains open.