Local weight-monodromy conjecture for nearby cycles

Let $K$ be a local field with residue characteristic $p$, let $\ell\neq p$ be a prime number, and let $\mathcal X$ be an admissible formal $\mathcal O_K$-scheme with smooth generic fiber $\mathcal X_\eta$. The nearby-cycle complex is

$$\rm{R}\Psi_{\mathcal X}\Q_\ell\in D^b_c(\mathcal X_s\times_s\eta;\Q_\ell).$$

A complex is monodromy-pure of weight zero when it has the monodromy-purity property of weight zero defined in the source. Local weight-monodromy conjecture. The nearby cycles $\rm{R}\Psi_{\mathcal X}\Q_\ell$ are monodromy-pure of weight zero. The conjecture concerns the local analogue of global weight-monodromy for formal schemes and their rigid-analytic generic fibers. In the paper, the subsequent theorem proves this assertion, so it is recorded here as solved.