Asymptotic expansion conjecture for extreme eigenvalues of non-Hermitian Toeplitz matrices

Let $\boldsymbol{T}_{n}(a)$ be the Toeplitz matrix generated by $a$, where $a$ satisfies properties $(i)$--$(iii)$ stated in the source, and let $\boldsymbol{ obreak\rm T}$ denote the unit circle. For a small positive number $\varepsilon$, consider the eigenvalues $lambda_{j}(T_{n}(a))$ with

$$j=1,\ldots,\lfloor\varepsilon n\rfloor$$

or

$$j=n-\lfloor\varepsilon n\rfloor+1,\ldots,n.$$

Extreme-eigenvalue expansion conjecture. There exists an integer $k\geqslant1$ such that

$$\lambda_{j}(T_{n}(a))=\sum_{\ell=0}^{k-1}\frac{q_{\ell}(j)}{(n+1)^{\alpha+\ell}}+E_{k,j,n},$$

where $|E_{k,j,n}|\leqslant c n^{-\alpha-k}$ for some constant $c$ depending only on $\alpha$, and, for every $\ell\geqslant1$, $q_{\ell}$ is a continuous function from the unit circle to $\mathbb{C}$ depending only on $a$. The conjecture proposes a higher-order asymptotic expansion of the extreme eigenvalues beyond the leading-order estimate, with a uniformly controlled remainder.