Frobenius semisimplicity conjecture for proper direct images of intersection complexes

Let $k$ be a finite field, let $f:X\to Y$ be a proper map of varieties over $k$, and let $y$ be a closed point of $Y$. Write $\overline{f^{-1}(y)}$ for the geometric fiber and $\overline{\mathcal{IC}_X}$ for the base change of the intersection complex. Frobenius semisimplicity conjecture. For every closed point $y$ of $Y$, the graded Galois module

$$H^*(\overline{f^{-1}(y)},\overline{\mathcal{IC}_X})$$

is semisimple. In particular, the direct image $f_*\mathcal{IC}_X$ is semisimple and Frobenius semisimple. This is presented as a fiberwise form of the paper's general conjecture and is proved in the source's stated special cases; no general resolution is supplied here.