The semisimple similarity number conjecture for matrix sequences

For $n,m\in\mathbb{N}$, let $\mathcal{S}_{n,m}$ be the subset of semisimple sequences in the space $V_{n,m}$ of $m\times m$ matrix sequences. Define the semisimple similarity number $S(m)$ by

$$S(m)=\min\{k\in\mathbb{N}:\ \forall n\in\mathbb{N},\ \forall\mathcal{A},\mathcal{B}\in\mathcal{S}_{n,m},\ \mathcal{A}\sim\mathcal{B}\Leftrightarrow\mathcal{A}_{J}\sim\mathcal{B}_{J}\ \forall J\text{ with }|J|\leq k\}.$$

Semisimple similarity number conjecture. For arbitrary $m\times m$ sequences, not necessarily semisimple, $\mathcal{A}\sim\mathcal{B}$ if and only if $\mathcal{A}_{J}\sim\mathcal{B}_{J}$ for every index vector $J$ with length at most $S(m)$.

The conjecture proposes that the number of terms needed to test simultaneous similarity for arbitrary matrix sequences is already controlled by the corresponding number for semisimple sequences. The source presents this as a proposed problem and gives no resolution.