Logarithmic Kollár vanishing conjecture for semistable reductions in positive characteristic

Let $X$ and $Y$ be proper and smooth $S$-schemes, let $f:X\rightarrow Y$ be an $E$-semistable $S$-morphism, and let $D$ be a simple normal crossing divisor on $X$ containing $X\times_Y E$. Let $H$ be a $\mathbb Q$-divisor on $X$ whose fractional part is supported on $D$ and such that $H\sim_{\mathbb Q}f^*L$ for an ample $\mathbb Q$-divisor $L$ on $Y$. Assume that $f:(X,D)\rightarrow(Y,E)$ has a lifting $\widetilde{f}:(\widetilde{X},\widetilde{D})\rightarrow(\widetilde{Y},\widetilde{E})$ over $\widetilde{S}$ and that $\dim(X/S)<p$. Logarithmic Kollár vanishing conjecture. Then

$$H^i\bigl(Y,R^jf_*\mathcal O_X(K_{X/S}+\ulcorner H\urcorner)\bigr)=0$$

for every $i>0$ and $j\geq 0$. This extends Kollár vanishing to semistable reductions with logarithmic and smooth horizontal coefficients in positive characteristic; the paper proves the corresponding decomposition and a positive-characteristic vanishing theorem in an integral-coefficient case, while this broader conjectural statement remains open.