Connectedness of closures of varieties of matrix tuples without irreducible tuples

Let ${\cal V}$ and ${\cal W}$ be the varieties of tuples of matrices considered in Theorem~, with the additive condition that the sum is zero or the multiplicative condition that the product is $I$, respectively. The tuples are taken from prescribed conjugacy classes, and no assumption is made that irreducible tuples exist.

Connectedness conjecture. The closures of ${\cal V}$ and ${\cal W}$ should be connected without the assumption that there are irreducible tuples.

All examples known to the author have connected closures. The source presents this as an interesting problem rather than as an established result.