Determinant of infinite Toeplitz matrices

Question

I have a question regarding determinants of Toeplitz matrices. In particular, let $A$ be a Toeplitz matrix with $m,n = 0,\dots, N-1$ non-negative integers with elements given by \begin{equation} A_{mn} = \int_{-\pi}^{\pi} \frac{dx}{2\pi} e^{i(m-n)x} A(x) \end{equation} with $A(x)$ a smooth function on the circle. Now consider the determinant $\det_{N}(A)$. Szegö's theorem then tells us the asymptotics of this determinant as $N$ goes to infinity, \begin{equation} \lim_{N\to \infty} \frac{\det_N(A)}{G(A)^{N+1}} = E(A) \end{equation} with \begin{equation} G(A) = \exp\left( \int_{-\pi}^\pi \frac{dx}{2\pi} \log A(x) \right),\quad E(A) = \exp\left( \sum_{k=1}^{\infty} \frac{1}{4\pi^2 k} \int_{-\pi}^\pi \int_{-\pi}^\pi dx dy \log A(x) \log A(y) e^{-i k (x-y)}\right). \end{equation} However, I am interested in a different determinant. Let $\widetilde{A}$ be a Toeplitz matrix given by \begin{equation} \widetilde{A}_{mn} = \int_{-\pi}^{\pi} \frac{dx}{2\pi} e^{i(m-n)x} \frac{1}{A(x)} \end{equation} again with $m,n = 0,\dots,N-1$. I now want to consider the determinant \begin{align} \det_{N}(\mathbb{1} + \alpha A\widetilde{A}) \end{align} with $\alpha$ a real constant. Are there known results for the asymptotic behaviour of these types of determinants with an explicit expression for the asymptotic behaviour, just like we have for Szegö's theorem? I had looked through the standard references and the book 'Analysis of Toeplitz matrices' by Böttcher and Silbermann, but couldn't find something useful. Even for the asymptotics of the determinant like $\det_N(A^2)$ I couldn't find a sharp statement. Any references/hidden gems would be greatly appreciated.