The l2 parametrization conjecture for absolutely continuous factorization measures

Let $\Sigma$ be the nondecreasing lift associated with the factorization map, and write $d\Sigma=\Sigma' d\theta$ when its measure is absolutely continuous with respect to Lebesgue measure. Let $[d\Sigma]=[d\theta]$ denote equality of measure classes, and let $\prod\Delta$ denote the product of unit disks. The l2 parametrization conjecture. One has

$$w\in l^2\quad\Longleftrightarrow\quad [d\Sigma]=[d\theta],$$

and equivalently there is a bijective correspondence

$$l^2\cap\prod\Delta\ \leftrightarrow\ \{\Sigma:d\Sigma=\Sigma' d\theta,\ \Sigma'>0\}.$$

This conjecture seeks a complete parameterization of the absolutely continuous, positive-density part of the factorization space by square-summable parameters; the source gives no resolution.