Simon's higher-order Szegő sum-rule conjecture

Let $w$ be the density of the absolutely continuous part of a measure on the unit circle, let $\alpha=(\alpha_n)_{n\geq 0}$ be its Verblunsky-coefficient sequence, and let $S$ be the left shift operator, $(S\alpha)_n=\alpha_{n+1}$. For distinct $\theta_1,\theta_2,\ldots,\theta_k\in[0,2\pi)$ and strictly positive integers $m_1,m_2,\ldots,m_k$, set $m=\max_{1\leq j\leq k}m_j$. Simon's higher-order Szegő conjecture. The condition

$$\int_{0}^{2\pi}\prod_{j=1}^{k}\bigl(1-\cos(\theta-\theta_j)\bigr)^{m_j}\log w(\theta)\frac{d\theta}{2\pi}>-\infty$$

should be equivalent to

$$\sum_{n=0}^{\infty}\left(\left|\left\{\prod_{j=1}^{k}\bigl(S-e^{-i\theta_j}\bigr)^{m_j}\alpha\right\}_n\right|^{2}+|\alpha_n|^{2m+2}\right)<\infty.$$

This conjecture was subsequently shown to be false: for $\theta_1=0$, $\theta_2=\pi$, and $m_1=m_2=1$, Lukic constructed a counterexample for which the sequence condition holds but the integral equals $-\infty$.