The crystalline companion conjecture for irreducible local systems

Let $S$ be the base scheme, let $X$ be the relevant smooth variety over $S$ with boundary divisors $D_i$, and let $T_i$ prescribe the monodromy at infinity. Suppose there is one, or infinitely many pairwise non-isomorphic, irreducible topological rank $r$ complex local system(s) $\mathbb L_{\mathbb C}$ with torsion determinant $\mathcal L$ and quasi-unipotent monodromies in $T_i$ at infinity. Crystalline companion conjecture. There is a non-empty open subscheme $S^\circ\subset S$ such that, for every closed point $s\in |S|$, there is one, or infinitely many pairwise non-isomorphic, Frobenius-invariant isocrystal(s) $M_{s'}$ on $X_{s'}$ (respectively, $M_{s_\alpha}$, each defined on some $X_{s_\alpha}$), where $s'\to s$ and $s_\alpha\to s$ are closed points, with determinant $\mathcal L$ as an $F$-isocrystal and residues modulo $\mathbb Z$ along $D_{i,\bar s}$ equal to the logarithms of the eigenvalues of $T_i$. This is the expected $\ell$-to-$p$ companion statement for Deligne's conjecture; the cited theorem would imply it if such companions exist, but the source presents the assertion as conjectural.