Nilpotence modulo 2 of the matrices associated with run-length encoding

Let $M_k(a,b)$ denote the matrix introduced in the paper for parameters $a,b,k$. The matrix $2M_k(a,b)$ is considered nilpotent modulo $2$ when some power of it has only even integer entries.

Nilpotence conjecture. For all $a\geqslant 0$, $b\geqslant 0$, and $k\geqslant 0$, there exists an integer $N\geqslant 0$ such that every entry of

$$\left(2M_k(a,b)\right)^N$$

is an even integer.

The claim is motivated by empirical evidence concerning the matrices arising from the paper's self-similar algebra constructions. Its resolution is not specified in the supplied text.