Full-measure conjecture for dominant products of matrix families

Let ${\cal{A}}=\{A_1,\ldots,A_m\}$ be a family of $m$ real $d\times d$ matrices, and let ${\cal{Q}}_m$ denote the space of all such families, identified with ${\mathbb R}^{m d^2}$. A product is dominant if, after normalizing the family so that its joint spectral radius is $1$, there exists $q<1$ such that every product not corresponding to a power of the dominant word or one of its cyclic permutations has spectral radius less than $q$. The leading eigenvalue is unique and simple when it is the only leading eigenvalue and has algebraic multiplicity one, with the complex case understood up to the conjugate pair. Full-measure conjecture. The set of families ${\cal{A}}\in{\cal{Q}}_m$ that possess dominant products with a unique and simple leading eigenvalue is of full Lebesgue measure in ${\cal{Q}}_m$. This would strengthen the known result that the set of such families is open and the genericity expectation that dominant products occur for almost every matrix family; the source gives no resolution of the full-measure assertion.