Lukic's decomposition conjecture for higher-order Szegő sum rules

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 $\theta_1,\ldots,\theta_k\in[0,2\pi)$ and strictly positive integers $m_1,\ldots,m_k$, let $m=\max_{1\leq j\leq k}m_j$. Lukic's conjecture. The finiteness condition

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

is equivalent to the existence of sequences $\beta^{(1)},\ldots,\beta^{(k)}$ such that

$$\alpha=\beta^{(1)}+\beta^{(2)}+\cdots+\beta^{(k)},$$

with

$$\bigl(S-e^{-i\theta_j}\bigr)^{m_j}\beta^{(j)}\in\ell^2$$

and

$$\beta^{(j)}\in\ell^{2m_j+2}$$

for every $j$. This refinement replaces the false pair of global conditions in Simon's conjecture and is presented as an improved conjecture.