The tensor-power null-summand conjecture for finite semigroups

Let $S$ be a finite semigroup without identity, let $k$ be a field, and let $V$ be a non-unital $kS$-module on which no element of $S$ acts invertibly. Let $Z$ denote the null representation, in which every element of $S$ acts as zero. The conjecture is

Null-summand conjecture. There is some $m\ge 1$ such that $Z$ is a direct summand of $V^{\otimes m}$, and consequently

$$a(n)=(\dim V)^n.$$

The preceding proposition establishes the analogous formula for the length growth, while this summand assertion is presented as the semigroup analogue of the earlier monoid conjecture. No resolution is supplied in the text.