Bogomolov–Pop's mod-$\ell$ isomorphism conjecture

Let $K|k$ and $L|l$ be function fields over algebraically closed fields. Let $\mathcal{G}_K^a$ and $\mathcal{G}_L^a$ denote the mod-$\ell$ abelianized Galois groups, and let $\underline{\operatorname{Isom}}^c(\mathcal{G}_L^a,\mathcal{G}_K^a)$ be the commutator-compatible isomorphisms modulo the natural $(\mathbb{Z}/\ell)^\times$-action. Let $\operatorname{Isom}^i_F(K,L)$ denote the Frobenius-preserving field isomorphisms, with the canonical map induced by the inclusion into the perfect closure. Bogomolov–Pop's mod-$\ell$ conjecture. If $\operatorname{char} K\neq\ell$ and $\operatorname{tr.deg}(K|k)\geq 2$, then the canonical map

$$\operatorname{Isom}^i_F(K,L)\longrightarrow\underline{\operatorname{Isom}}^c(\mathcal{G}_L^a,\mathcal{G}_K^a)$$

is a bijection. This is the mod-$\ell$ analogue of the pro-$\ell$ isomorphism conjecture and concerns reconstruction of function fields from their abelianized mod-$\ell$ Galois data. The supplied source gives no resolution evidence, so the conjecture is recorded as open.