The strong nilpotency index conjecture over reduced noncommutative rings

Let $M$ be a square matrix of size $m$ whose entries are functions from a set $S$ to a reduced noncommutative ring $R$. For $v\in S$, write $M|_v$ for the matrix obtained by evaluating the entries of $M$ at $v$. Suppose that there exists an $r\in\mathbb{N}$ such that

$$M|_{v_r}M|_{v_{r-1}}\cdots M|_{v_1}=0$$

for all $v_i\in S$. The strong nilpotency index conjecture. Then

$$M|_{v_m}M|_{v_{m-1}}\cdots M|_{v_1}=0$$

for all $v_i\in S$. The analogous assertion is proved in the paper for reduced commutative rings, while the noncommutative case is presented as unproved because domains need not embed in division rings.