Uniform asymptotic expansion conjecture for eigenvalues of higher-order Toeplitz symbols

Let $g_m(x)=\left(2\sin\frac{x}{2}\right)^{2m}$ for an integer $m\geq 3$, and let $\lambda_{n,j}$ be the eigenvalues of the Toeplitz matrices generated by $g_m$. Let $d_k$ denote the coefficient functions in the proposed regular expansion. Higher-order Toeplitz expansion conjecture. If $p\leq 2m-1$, there exist $N_p\in\mathbb{N}$ and $D_p>0$ such that

$$\left|\lambda_{n,j}-\sum_{k=0}^p \frac{d_k\left(\frac{j\pi}{n+2}\right)}{(n+2)^k}\right|\leq\frac{D_p}{(n+2)^{p+1}}$$

for all $n\geq N_p$ and all $j\in\{1,\ldots,n\}$. For $p=2m$, this inequality does not hold for all sufficiently large $n$ and all $1\leq j\leq n$, but it does hold for all sufficiently large $n$ and $(\log(n+2))^2\leq j\leq n$. This predicts uniform regular asymptotics through order $2m-1$, while identifying the boundary layer near $j=1$ as the obstruction at order $2m$.