Variational filtered convergent log hard Lefschetz conjecture

Let $X/S$ be a proper SNCL scheme in characteristic $p>0$, with relative dimension $d$ over $S_1$, and let $L$ be a relatively ample line bundle on $X/S_1$. Let $c_{1,\operatorname{conv}}$ be the convergent first Chern class and set

$$\eta:=c_{1,\operatorname{conv}}(L).$$

Here $(i)$ denotes the twist whose filtration is defined by

$$P_k\bigl(R^{d+i}f_{X/S*}({\cal K}_{X/S})(i)\bigr)=P_{k+2i}R^{d+i}f_{X/S*}({\cal K}_{X/S}).$$

Variational filtered convergent log hard Lefschetz conjecture. For the relevant integers $i$, the cup product

$$\eta^i\colon R^{d-i}f_{X/S*}({\cal K}_{X/S})\longrightarrow R^{d+i}f_{X/S*}({\cal K}_{X/S})(i)$$

is an isomorphism; moreover, it is an isomorphism of filtered sheaves

$$\eta^i\colon \bigl(R^{d-i}f_{X/S*}({\cal K}_{X/S}),P\bigr)\xrightarrow{\sim}\bigl(R^{d+i}f_{X/S*}({\cal K}_{X/S})(i),P\bigr).$$

This is the filtered convergent logarithmic analogue of hard Lefschetz in the variational setting. The source states it as a conjecture and gives no evidence of resolution.