Esnault–Langer conjecture on compatible points of Frobenius twists of an abelian variety

Let $L$ be a field finitely generated over an algebraically closed field $l_0$ of characteristic $p$, and let $C$ be an abelian variety over $L$. For each $\ell\geqslant 0$, choose a point $x_\ell\in C^{(p^\ell)}(L)$, and for each $\ell\geqslant 1$ let $V_{C^{(p^\ell)}/L}$ denote the Verschiebung relative to $L$. Write $\operatorname{Tr}_{L|l_0}(C)$ for the $L|l_0$-trace of $C$. Esnault–Langer conjecture. If

$$V_{C^{(p^\ell)}/L}(x_\ell)=x_{\ell-1}$$

for every $\ell\geqslant 1$, then the image of $x_0$ in

$$C(L)/\operatorname{Tr}_{L|l_0}(C)(l_0)$$

is a torsion point of order prime to $p$. This concerns the prime-to-$p$ torsion remaining after quotienting by the constant part of the abelian variety; the supplied source does not establish the assertion's status.