Filtered log convergent hard Lefschetz conjecture

Let $f\colon X\to S$ be in the log convergent setting of the paper, with relative dimension $d$, and let $P$ denote the weight filtration. Let $N$ be the monodromy parameter and assume $N=0$. Let $\eta$ be the cup-product class of a relatively ample line bundle. Filtered log convergent hard Lefschetz conjecture. The cup product

$$\eta^i\colon R^{d-i}f_*({\cal O}_{X/K})\longrightarrow R^{d+i}f_*({\cal O}_{X/K})(i),\qquad i\in\mathbb N,$$

is an isomorphism in $F\operatorname{\textrm{-}Isoc}^{\square}(S/{\cal V})$. In fact, it is an isomorphism of filtered sheaves

$$\eta^i\colon (R^{d-i}f_*({\cal O}_{X/K}),P)\overset{\sim}{\longrightarrow}(R^{d+i}f_*({\cal O}_{X/K})(i),P),$$

where $P_k(R^{d+i}f_*({\cal O}_{X/K})(i)):=P_{k+2i}R^{d+i}f_*({\cal O}_{X/K})$. This is the log convergent counterpart of the filtered log crystalline hard Lefschetz conjecture; the supplied text gives no resolution evidence.