Modified higher-order Szegő conjecture for orthogonal polynomials on the unit circle

Let $\mu$ be a probability measure on the unit circle with infinite support, write

$$d\mu=w(\theta)\frac{d\theta}{2\pi}+d\mu_{\mathrm{s}},$$

and let $\alpha=\{\alpha_n\}_{n=0}^{\infty}$ be its Verblunsky coefficients. Let $S$ be the shift on sequences, $(Sx)_n=x_{n+1}$. For $m_1,\dots,m_l\in\mathbb{N}$ and distinct $\theta_1,\dots,\theta_l\in[0,2\pi)$, define the conditions

$$\alpha=\beta^{(1)}+\cdots+\beta^{(l)},$$ $$(S-e^{-i\theta_k})^{m_k}\beta^{(k)}\in\ell^2,$$

and

$$\beta^{(k)}\in\ell^{2m_k+2}\quad\text{for all }k.$$

Modified higher-order Szegő conjecture. The weighted logarithmic integrability condition

$$\int\prod_{k=1}^l(1-\cos(\theta-\theta_k))^{m_k}\log w(\theta)\frac{d\theta}{2\pi}>-\infty$$

is equivalent to the existence of sequences $\beta^{(1)},\dots,\beta^{(l)}$ satisfying all three displayed conditions.