GLT closure under geometric mean

Let $\{A_n\}_n$ and $\{B_n\}_n$ be Hermitian positive definite GLT matrix-sequences with matrix-valued GLT symbols

$$\{A_n\}_n\sim_{\mathrm{GLT}}\kappa,\qquad \{B_n\}_n\sim_{\mathrm{GLT}}\xi.$$

Assume $\kappa,\xi:[0,1]^d\times[-\pi,\pi]^d\to\mathbb{C}^{r\times r}$ are positive-semidefinite. For $\varepsilon>0$, set $\kappa_\varepsilon=\kappa+\varepsilon I_r$ and $\xi_\varepsilon=\xi+\varepsilon I_r$, and define

$$\widetilde G(\kappa,\xi)(x,\theta):=\lim_{\varepsilon\to 0}G\bigl(\kappa_\varepsilon(x,\theta),\xi_\varepsilon(x,\theta)\bigr)$$

for almost every $(x,\theta)$. GLT closure under geometric mean. The geometric mean matrix-sequence is again GLT and

$$\{G(A_n,B_n)\}_n\sim_{\operatorname{GLT}}\widetilde G(\kappa,\xi).$$

Here $G$ denotes the matrix geometric mean, and $\widetilde G(\kappa,\xi)$ is the candidate symbol defined by the preceding limit. This conjecture seeks to remove invertibility assumptions on the GLT symbols from spectral-distribution results for geometric means of HPD GLT sequences; its status is not resolved by the supplied text.