The Hilbert class field tower conjecture for curves over finite fields

Let $X$ and $Y$ be smooth projective curves of genus at least two over a finite field ${\mathbb F}_q$. Let $H_0(X):=X$, let $H_1(X):=H(X)$, and recursively let $H_{n+1}(X):=H_n(H(X))$ for $n\geq 1$, where $H(X)$ is the unramified abelian cover defined by the pullback of $X$ under $I-\operatorname{Fr}:J_X\to J_X$, with twists determined by a choice of degree-one divisor embedding. Hilbert class field tower conjecture. If, for each $n$, there are choices of twists such that the $L$-function of $H_n(X)$ equals the $L$-function of $H_n(Y)$ for all $n\geq 0$, then $X$ is isomorphic to $Y$. The conjecture proposes that the $L$-functions of the successive Hilbert class field tower covers determine the curve, strengthening the information obtained from the Jacobian and its Tate-module Galois representation. The supplied text gives no resolution, so the status remains open.