Low-rank perturbation conjecture for Toeplitz spectral measures

Let $A$ satisfy the conditions of Theorem 3.1, and let $A'$ be the matrix constructed in Lemma 3.4. For the Toeplitz coefficients $a_j$, define the limiting measure $\mu$ as the distribution of

$$\sum_{|j|\leq k}a_jU^j,$$

where $U$ is uniformly distributed on $S^1$. Low-rank perturbation conjecture. The empirical spectral measure $\mu_{A+n^{-\gamma}A'}$ converges weakly in probability to $\mu$. This is suggested by the proof of the cited perturbation theorem, but is not established there and remains open.