Generalized weight-monodromy conjecture for rigid-analytic varieties with projective reduction

Let $K$ be a $p$-adic local field with residue field of cardinality $q$, let $X$ be a smooth proper rigid-analytic $K$-variety, let $\ell\neq p$ be a prime number, and suppose that $X$ admits an admissible formal $\mathcal O_K$-model $\mathcal X$ with projective special fiber $\mathcal X_s$. Let $\bar\eta$ denote a geometric generic point, and let $\operatorname{Fil}_{\mathrm{M}}^\bullet$ be the monodromy filtration on $\rm{H}^i(X_{\bar\eta},\Q_\ell)$. Generalized weight-monodromy conjecture. The eigenvalues of any geometric Frobenius lift on

$$\operatorname{gr}^j_{\mathrm{M}} \rm{H}^i(X_{\bar\eta},\Q_\ell)$$

are $q$-Weil numbers of weight $i+j$ for every pair of integers $i,j$. This generalizes the weight-monodromy prediction from algebraic varieties to rigid-analytic varieties admitting a model with projective special fiber. The paper presents it as a suggested generalization and explains that a positive answer to the stated projective direct-image question would imply it; the general claim remains open.