Determinantal process conjecture for zeros of Gaussian matrix-valued power series

Let $A_k$, for $k\geq 0$, be independent $n\times n$ matrices whose entries are independent standard complex Gaussian random variables. The zeros in the unit disk of the determinant of the matrix-valued Gaussian power series

$$A_0+zA_1+z^2A_2+\cdots$$

form a determinantal point process on $\mathbb D$ with kernel

$$\mathbb K(z,w)=\genfrac{}{}{}{}{n}{\pi}\genfrac{}{}{}{}{(1-|z|^2)^{(n-1)/2}(1-|w|^2)^{(n-1)/2}}{(1-z\overline{w})^{n+1}}.$$

Here the reference measure is Lebesgue measure on $\mathbb D$. Equivalently, $\mathbb K$ is the projection kernel onto the subspace of analytic functions in $L^2\left(\mathbb D,\genfrac{}{}{}{}{n}{\pi}(1-|z|^2)^{n-1}\,dm(z)\right)$. Determinantal process conjecture. The asserted determinantal-process description and kernel hold.