The l2 sufficient condition for factorization limits to be homeomorphisms

Fix a permutation $p$ of the natural numbers, and let $\sigma_N$ be the associated finite factorization maps with limit $\sigma(p,w)$, where $w=(w_n)$ belongs to the product of unit disks. Let $l^2$ denote the square-summable sequences. The l2 homeomorphism conjecture. If $w\in l^2$, then

$$\sigma(p,w):=\lim_{N\to\infty}\sigma_N\in \operatorname{Homeo}(S^1).$$

The conjecture proposes a sufficient condition for the limiting degree-one circle map to be invertible in the less regular setting; the source gives no resolution.