The absolute continuity dichotomy for factorization measures

Fix a permutation $p$ and write $\sigma(p,w)(e^{i\theta})=e^{i\Sigma(\theta)}$, where $\Sigma$ is the associated nondecreasing lift. Let $d\Sigma/(2\pi)=\Sigma'(\theta)d\theta/(2\pi)+\mu_s$ be its decomposition into absolutely continuous and singular parts, and let $l^2$ be the space of square-summable parameter sequences. The absolute continuity dichotomy conjecture. (a) If $w\in l^2\cap\prod_{n=1}^{\infty}\Delta$, then

$$\Sigma'(\theta)=\left(\prod_{n=1}^{\infty}(1-|w_n|^2)\right)\exp\left(-2\sum_{n=0}^{\infty}\operatorname{Re}\left(\log(1+w_{n'}\sigma_{n-1}(z)^{n'})\right)\right),$$

where the partial sums converge in $L^2(d\Sigma)$. (b) If $w\in l^2$, then $\Sigma'>0$ almost everywhere with respect to $d\theta$. (c) If $w\notin l^2$, then $\Sigma'=0$ almost everywhere with respect to $d\theta$, equivalently $d\Sigma\perp d\theta$. The claim is intended to distinguish absolutely continuous and singular behavior of the limiting factorization map; the source gives no resolution.