Monodromy conjecture for log mixed objects

Let $k$ be the base field, let ${\mathcal {P}}$ be the class of schemes considered in the source, and let $X\in {\mathcal {P}}$. The monodromy operator is

$$N:H^m(X)_{\ell}\to H^m(X)_{\ell}(-1).$$

The associated objects $H^m(X)^{\flat}$, $H^m(X)(-1)^{\flat}$, $H^m(X)^{\flat*}$, and $H^m(X)(-1)^{\flat*}$ are defined through the log and enlarged log categories. Monodromy conjecture. The operator $N$ comes from geometry rather than only from Galois theory: it is a morphism

$$H^m(X)^{\flat}\to H^m(X)(-1)^{\flat},$$

and hence a morphism

$$H^m(X)^{\flat*}\to H^m(X)(-1)^{\flat*}.$$

This is part of the broader comparison conjectures relating motivic morphisms to Hodge- and Tate-theoretic morphisms. The supplied text records it as an expectation and gives no resolution evidence.