The v-adic period relation lifting conjecture for t-motives

Let $K:=\mathbb{F}_{q}(\theta)$ be the rational function field, let $K_{v(\theta)}$ be its completion at the place $v(\theta)$, and let $K(t)_{v}$ and $K^{\mathrm{sep}}[t]_{v}$ be the corresponding $v$-adic rings. Let $\Phi$ and $\psi$ satisfy

$$\Phi \in \operatorname{GL}_{r}(K(t)_{v}) \cap \operatorname{Mat}_{r \times r}(K[t]),\qquad \psi \in \operatorname{Mat}_{r \times 1}(K^{\mathrm{sep}}[t]_{v}),$$

with $\psi(\theta)$ convergent, $\sigma\psi=\Phi\psi$, and $\det\Phi=c(t-\theta)^{s}$ for some $c\in K^{\times}$ and $s\in\mathbb{N}$. The $v$-adic period relation lifting conjecture. If $\rho\in\operatorname{Mat}_{1\times r}(K_{v(\theta)})$ satisfies

$$\rho\psi(\theta)=0,$$

then there exists $P\in\operatorname{Mat}_{1\times r}(K[t]_{v})$ such that

$$P\psi=0,\qquad P(\theta)\text{ converges},\qquad P(\theta)=\rho.$$

Thus every linear relation among the components of $\psi(\theta)$ over $K_{v(\theta)}$ lifts to a linear relation among the components of $\psi$ over $K[t]_{v}$. This is presented as a $v$-adic analogue of Proposition 3.1.1 in the Anderson--Brownawell--Papanikolas theory; the supplied text gives no resolution, so the conjecture remains open.