Asymptotic change-of-variable conjecture for preconditioned Toeplitz eigenvalues

Let $l,g$ be two even functions with $g>0$ on $(0,\pi)$, and suppose that $f\equiv l/g$ is monotone increasing over $(0,\pi)$. Set $X_n\equiv T_n^{-1}(g)T_n(l)$ for all $n$. Let $s_{j,n}\in(0,\pi)$ satisfy $\lambda_j(X_n)=f(s_{j,n})$, let $h\equiv 1/(n+1)$, and set $\theta_{j,n}\equiv \pi jh$. Asymptotic change-of-variable conjecture. For some integer $K\geq 0$, every $n$, and every $j=1,\ldots,n$, the points $s_{j,n}$ have the expansion

$$s_{j,n}=\theta_{j,n}+\sum_{k=1}^{K}\rho_k(\theta_{j,n})h^k+E_{j,n,K},$$

where the $s_{j,n}$ are arranged in nondecreasing order, each $\rho_k:(0,\pi)\to\mathbb{R}$ is continuous and depends only on $f$, and $E_{j,n,K}=O(h^{K+1})$ with $|E_{j,n,K}|\leq c h^{K+1}$ for a constant $c$ depending only on $K,l,g$. The result would refine the known zeroth-order localization $s_{j,n}=\theta_{j,n}+o(1)$ and underlies more precise matrix-less algorithms. Its validity for the preconditioned setting remains open.