Shiho's stronger conjecture on framewise relative cohomology

Suppose that $(X,\overline{X})\to(S,\overline{S})$ is a proper Cartesian morphism of $k$-pairs with $X\to S$ smooth. Let $E$ be an overconvergent $(F)$-isocrystal on $(X,\overline{X})/K$, and let $q\geq0$. For every frame $(T,\overline{T},\mathfrak{T})$ over $(S,\overline{S})$, with $\mathfrak{T}$ smooth over $\mathcal{V}$ near $T$, the framewise rigid direct image satisfies compatibility with natural base change morphisms.

Stronger Shiho conjecture. There exists a unique overconvergent $(F)$-isocrystal $\widetilde E$ on $(S,\overline{S})$ such that

$$\widetilde E_{(T,\overline{T},\mathfrak{T})}\cong\mathbf{R}^qf'_{\mathfrak{T},\mathrm{rig}*}E|_{(X_T,\overline{X}_{\overline{T}})}.$$

The transition morphisms are the natural base change morphisms; if $\overline S$ is proper, $\widetilde E$ depends only on $f:X\to S$ and $E$.

This is explicitly described as a stronger version of Shiho's conjecture. The source does not provide a resolution.