The torsion conjecture for infinitely $p$-divisible points on abelian varieties

Let $K_0$ be the function field of a smooth curve over a finite field ${\Bbb F}$ of characteristic $p$, let $K_0^{\rm s}$ be a separable closure of $K_0$, and let $B$ be an abelian variety over $K_0$. For an abelian group $G$, define

$$G^\#:=\bigcap_{l\geqslant 0}p^lG.$$

The elements of $G^\#$ are called infinitely $p$-divisible points, and let ${\rm Tor}(G)$ denote the subset of elements of finite order. Torsion conjecture. The inclusion

$$B(K_0^{\rm s})^\#\subseteq{\rm Tor}(B(K_0^{\rm s}))$$

holds. This conjecture asks whether every infinitely $p$-divisible point on an abelian variety over the separable closure of such a function field is torsion; it was raised by Benoist, Bouscaren and Pillay and is presented here without a resolution.