The conjecture on the minimum eigenvalue derivative for block-Toeplitz symbols

Let $\deg>0$, $j>0$, and $z>0$. Let $\hat{f}_{z,j}$ be the symbol appearing in the paper, and let $\lambda”_{\rm \min}(\hat{f}_{z,j})|_{0}$ denote the second derivative at $0$ of its minimum eigenvalue. The minimum-eigenvalue derivative conjecture. There exists a constant $c_{z,\deg}>0$ such that

$$\left.\lambda”_{\rm \min}(\hat{f}_{z,j})\right|_{0}=c_{z,\deg}\left(\frac{z^2}{2}\right)^j.$$

This conjecture is used to extend the condition-number analysis from polynomial degree $2$ to higher degrees; it was numerically verified in the paper for $\deg=3,4$, but no general proof is supplied.