Almost-sure or probabilistic mixing of independent separable Brownian permutons

Determine whether, for an i.i.d. sequence of independent biased Brownian separable permutons $P_1,P_2,\ldots$, the products $P_1\cdots P_k$ converge to the uniform permuton $U$ almost surely or in probability as $k\to\infty.

Background

The paper proves only convergence of the expectations $E[P_1\cdots P_k]$ to the uniform permuton $U$. It explicitly raises the stronger unresolved question of whether the random products themselves converge to $U$ almost surely or in probability. Such a result would mean that the structural effects of separable permutations disappear under sufficiently many independent compositions, not merely after averaging.