Rigid Clemens–Schmidt conjecture for limiting Frobenius structures

Let $X\to S\subseteq \mathbb A^1_{\mathbb F_q}\setminus\{0\}$ be a morphism of varieties over $\mathbb F_q$ with a good lift around $t=0$. Let $({\mathcal H}_0^m,{\mathcal N}_0,{\mathcal F}_0,e_m)$ be the limiting Frobenius structure at $t=0$ in dimension $m$, and define

$$e=\operatorname{lcm}_{0\leq m\leq 2n}(e_m).$$

Let $X_e\to S_e$ be obtained by pulling back along $S_e\to S$. Rigid Clemens–Schmidt conjecture. The morphism $X_e\to S_e$ extends to a semistable degeneration with degenerate fibre $X_{e,0}$ and, for every such extension, there exists a covariant rigid homology functor $H_{j,\operatorname{rig}}(\bullet)$ on semistable $\mathbb F_q$-varieties, vanishing outside $0\leq j\leq 2n$, for which an exact sequence of $F$-isocrystals contains

$$\cdots\to H_{2n+2-m,\operatorname{rig}}(X_{e,0})(-n-1)\to H^m_{\operatorname{rig}}(X_{e,0})\to {\mathcal H}_0^m\xrightarrow{{\mathcal N}_0}{\mathcal H}_0^m(-1)$$ $$\to H_{2n-m,\operatorname{rig}}(X_{e,0})(-n-1)\to H^{m+2}_{\operatorname{rig}}(X_{e,0})\to {\mathcal H}_0^{m+2}\xrightarrow{{\mathcal N}_0}{\mathcal H}_0^{m+2}(-1)\to\cdots.$$

This is the rigid-cohomological analogue of the Clemens–Schmidt sequence for complex semistable degenerations and would give the limiting Frobenius structure the predicted geometric meaning.