The hyperelliptic Jacobian torsion conjecture

Let $K$ be a number field, let $g$ be the genus of a hyperelliptic curve, and let $p$ be the prime occurring in the modular-tower construction. Let $G_K$ be the absolute Galois group of $K$, and let $C\cong\mathbb Z/p^{k+1}$ be a cyclic subgroup of torsion on a hyperelliptic Jacobian of genus $g$.

Hyperelliptic Jacobian torsion conjecture. For sufficiently large $k$, there is no such $C$ for which $G_K$ acts on $C$ through its cyclotomic action on $\langle e^{2\pi i/p^{k+1}}\rangle$.

The source states this as the hyperelliptic-Jacobian formulation equivalent to the weak conjecture for the indicated dihedral modular tower. No resolution is supplied in the excerpt.