Sutherland–Voloch anabelian conjecture for Hilbert class field towers of curves

Let $C$ and $D$ be smooth projective curves of equal genus at least $2$ over a finite field ${\mathbb F}$. Embed $C$ into its Jacobian and define its Hilbert class field cover $H(C)$; set $H_0(C):=C$, $H_1(C):=H(C)$, and recursively $H_{n+1}(C):=H_n(H(C))$ for $n\geq 1$, with analogous definitions for $D$. Sutherland–Voloch conjecture. If, for each $n$, there are choices of twists such that the $L$-function of $H_n(C)$ equals the $L$-function of $H_n(D)$ for all $n\geq 0$, then $C$ is isomorphic to a conjugate of $D$. This conjecture relates arithmetic data from iterated Hilbert class field covers to the isomorphism class of a curve; the supplied text does not state a resolution.