He–et al.'s summand-growth conjecture for finite monoids

Let $M$ be a finite monoid with group of units $G$, let $k$ be a field, and let $V$ be a $kM$-module on which no element apart from $1\in M$ acts as the identity. For each $n$, let $a(n)$ denote the relevant growth sequence for the restriction of $V$ to $G$, let $b(n)$ denote the number of summands of $V^{\otimes n}$, let $Z_V(G)$ be the subset of conjugacy-class representatives $g_t$ appearing in the stated character formula, let $\omega_V(g_t)$ be the associated eigenvalue, and let $S_t$ be the sum over the conjugates of the $t$-th column of the simple Brauer character table of $G$. The conjectured formula is

He–et al.'s conjecture. There is some $n\in\mathbb{N}$ such that $V^{\otimes n}$ contains a direct summand of the form $\operatorname{Ind}(W)$ for a $kG$-module $W$.

This conjecture is proposed as a sufficient condition for the asserted asymptotic formula for $b(n)$, namely $b(n)\sim a(n)$ with the character expression given in the source. The supplied text gives no resolution of the summand assertion.