Non-torsion primitive-point conjecture for large almost simple Galois extensions

Let $C/K$ be a smooth, projective, hyperelliptic curve of genus $g\geq 1$ with a $K$-rational point $P_0$, and embed $C$ into its Jacobian $J_C$ using $P_0$. Let $L/K$ be a finite Galois extension, write $G=\operatorname{Gal}(L/K)$, and let $P\in C(L)$ be $L$-primitive if $L$ is the smallest Galois extension of $K$ over which $P$ is defined. Let $m(G)$ denote the quantity used in the source. The non-torsion primitive-point conjecture. If $G$ is nonabelian almost simple and $|G|$ is sufficiently large, then $P$ is non-torsion in $J_C(L)$; in particular,

$$\dim J_C(L)\otimes\mathbb Q\geq m(G).$$

The conjecture connects large nonabelian Galois groups of fields of definition with Jacobian rank; it is explicitly described as very optimistic and remains open in the source.