Almost-sure welding and triangular-factorization conjecture

Let $\mu_0$ be Werner's measure on self-avoiding loops, let $\nu_0$ be its induced measure on welding homeomorphisms $\sigma$, and let $W$ be the welding map. A quasicircle is a Jordan curve admitting a parametrization obtained by restricting a quasiconformal homeomorphism of $\mathbb C\cup\{\infty\}$ to $S^1$. A triangular factorization is a factorization $\sigma=lau$ with the middle rotational factor normalized by $m=1$. Welding-factorization conjecture. (a) $\mu_0$ has measure zero on quasicircles; (b) $W$ is one-to-one on a set of full $\mu_0$ measure; and (c) $\nu_0$-almost surely, $\sigma$ has a unique triangular factorization with $m=1$. These are unresolved foundational issues needed to make the welding parametrization and factorization description rigorous almost surely.