Nonsingularity of a twisted Toeplitz matrix

Question

Let $A=(a_{p-q})_{p,q=1}^n$ be an $n\times n$ Toeplitz matrix, that is, $$A=\begin{pmatrix}a_0 & a_{-1} & a_{-2} & \cdots & a_{1-n} \\ a_1 & a_0 & a_{-1} & \cdots & a_{2-n} \\ a_2 & a_1 & a_0 & \cdots & a_{3-n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n-1} & a_{n-2} & a_{n-3} & \cdots & a_0\end{pmatrix}$$ Criteria for non-singularity of such matrices have been proven by Gohberg–Semencul, Gohberg–Krupnik, Lv–Huang and others. If $A$ has entries in some field $K$, and $\sigma$ is an automorphism of $K$ of order $n$, then we can consider the following $\sigma$-twist of $A$: $$A^\sigma=\begin{pmatrix}a_0 & a_{-1} & a_{-2} & \cdots & a_{1-n} \\ \sigma(a_1) & \sigma(a_0) & \sigma(a_{-1}) & \cdots & \sigma(a_{2-n}) \\ \sigma^2(a_2) & \sigma^2(a_1) & \sigma^2(a_0) & \cdots & \sigma^2(a_{3-n}) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \sigma^{n-1}(a_{n-1}) & \sigma^{n-1}(a_{n-2}) & \sigma^{n-1}(a_{n-3}) & \cdots & \sigma^{n-1}(a_0)\end{pmatrix}$$ Does there exist a similar criterion for checking nonsingularity of $A^\sigma$?