The local invariant cycle theorem for regular schemes

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 $X_{\bar{s}}$ and $X_{\bar{\eta}}$ for its geometric special and generic fibres, and let

$$\operatorname{sp}:H^i(X_{\bar{s}},\mathbb{Q}_l)\longrightarrow H^i(X_{\bar{\eta}},\mathbb{Q}_l)^I$$

be the specialization morphism, where $I$ is the inertia subgroup and $l\neq p$. Let $W_j$ denote the weight filtration on both sides. Local invariant cycle theorem. If $X$ is regular, then $\operatorname{sp}$ is an epimorphism and induces an isomorphism on $W_{i-1}$. This is the $l$-adic local invariant cycle statement, analogous to the proposed $p$-adic version; the source provides no resolution status for this formulation.