Pro-l Galois-group isomorphism conjecture for the fields of Theorem 2

Let $\ell$ be a prime, and let $K$ and $L$ be any fields as in Theorem 2 of the source paper. Write $\mathcal{G}_{K}(\ell)$ for the relevant pro-$\ell$ Galois group. Pro-$\ell$ Galois-group isomorphism conjecture. The groups associated with $K$ and $L$ should be isomorphic as topological groups:

$$\mathcal{G}_{K}(\ell) \simeq_{\textup{top.gr.}} \mathcal{G}_{L}(\ell).$$

This conjecture extends the proposed indistinguishability of the relevant fields from their pro-nilpotent Galois groups to their pro-$\ell$ Galois groups. The supplied excerpt does not reproduce the hypotheses of Theorem 2 or provide evidence of resolution, so the conjecture remains open.