The weight-zero $p$-adic invariant-cycles conjecture

Let $X$ be regular, let $W_0$ be the sum of generalized $\phi$-eigenspaces whose eigenvalues are roots of unity, and let $\operatorname{sp}$ be the étale specialization map. Weight-zero invariant-cycles conjecture. The map

$$W_0H^i(X_{\bar s},\mathbb Q_p)\xrightarrow{\operatorname{sp}}W_0H^i(X_{\bar\eta},\mathbb Q_p)^I$$

is an isomorphism. The source records this as the portion of the preceding $p$-adic invariant-cycles statement needed for its applications, namely the weight-zero (and smaller eigenvalue-one) part.