Questions on negative indices of rigidity in the Deligne–Simpson problem

Let $(p+1)$-tuples of matrices $M_j$ or $A_j$ have a specified index of rigidity, and let the expected dimension refer to the dimension predicted for the variety of all such $(p+1)$-tuples. A tuple has a non-trivial centralizer when its common centralizer is larger than the scalar matrices. Let $J^n_1,\ldots,J^n_{p+1}$ be Jordan normal forms, and suppose they satisfy the conditions of Theorem~.

Questions on negative indices of rigidity. 1) Is the dimension of the variety of $(p+1)$-tuples with non-trivial centralizers always smaller than the expected dimension of the variety of all $(p+1)$-tuples of matrices $M_j$ or $A_j$? 2) If the index of rigidity is negative and the Jordan normal forms $J^n_1,\ldots,J^n_{p+1}$ satisfy the conditions of Theorem~, is the Deligne–Simpson problem weakly solvable for any eigenvalues?

These questions arise from the dimension count for varieties of tuples with non-trivial centralizers and from the preceding example with zero index of rigidity. The source does not provide a resolution of either question.