Weak convergence conjecture for the pro-ell random-group measures

Let $\Gamma=\mathbb{Z}/2\mathbb{Z}$ and let $ell$ be an odd prime. Let ${\mathcal P}_{\Gamma,q,\ell}$ be the coarser topological space obtained by restricting the basic opens of ${\mathcal P}_{\Gamma,q}$ to finite $ell$-group quotients, and let $nu_{q,\ell,n}$ be the probability measures defined from the random groups $Y_{q,n}$.

Weak convergence conjecture. On ${\mathcal P}_{\Gamma,q,\ell}$, the measures $nu_{q,\ell,n}$ converge weakly, as $n\to\infty$, to the measure $mu_{\Gamma,q}$ from the existence and uniqueness conjecture.

The conjecture asserts that the pro-$\ell$ random-group model realizes the predicted limiting distribution in the case $\Gamma=\mathbb{Z}/2\mathbb{Z}$. It is a refinement of the moment calculation for $Y_{q,n}$ and remains open.