Matrix-algebra conjecture for simple and quasi-simple automata

Let $\mathcal{A}$ be either a simple or quasi-simple automaton with $n$ states, and let $\mathcal{R}(\mathcal{A})$ be its associated synchronized $\mathbb{C}$-algebra. Matrix-algebra conjecture. There exists an integer $m\leq n-1$ such that

$$\mathcal{R}(\mathcal{A})\cong\mathbb{M}_{m}(\mathbb{C}).$$

The conjecture is motivated by the computation that the algebra associated with the Černý automata is a full matrix algebra, while the corresponding result for all simple automata is missing. The supplied text gives no resolution.