Dense spectral finiteness for constrained matrix tuples

Let $\mathbb{A}\in\{0,1\}^{K\times K}$ be any matrix, and let

$$\mathscr{M}_K^{d\times d}=\left\{\boldsymbol{A}=(A_1,\dotsc,A_K)\mid A_k\in\mathbb{R}^{d\times d}\textrm{ for }1\leq k\leq K\textrm{ and }\boldsymbol{\rho}(\boldsymbol{A},\mathbb{A})=1\right\}$$

with its natural subspace topology. Here spectral finiteness restricted to $\mathbb{A}$ means that the joint spectral radius is attained by a periodic product allowed by the transition matrix $\mathbb{A}$. Dense Spectral Finiteness conjecture. The elements $\boldsymbol{A}\in\mathscr{M}_K^{d\times d}$ having spectral finiteness restricted to $\mathbb{A}$ should be dense in $\mathscr{M}_K^{d\times d}$. This proposes that constrained spectral finiteness is a generic approximation property on the normalized space of matrix tuples. The statement is posed as an open question; the supplied text gives no resolution.