The monodromy-weight conjecture for projective log smooth morphisms

Let $f:X\to Y$ be a projective log smooth vertical saturated morphism of fs log schemes over $k$. The direct image $R^mf_*(\overline{\mathbb{Q}}_{\ell})$ is an object of the category ${\mathcal A}_Y$ of pure $W$-weight $m$.

Monodromy-weight conjecture. For every integer $m$, $R^mf_*(\overline{\mathbb{Q}}_{\ell})$ is an object of ${\mathcal A}_Y$ of pure $W$-weight $m$.

This conjecture relates the category ${\mathcal A}_X$ to the monodromy-weight conjecture. The paper describes ${\mathcal A}_X$ as an analogue of the category of variations of mixed Hodge structure and notes that it has related asymptotic applications, but the supplied text gives no evidence that this assertion has been resolved.