Cartan nonsingularity conjecture for monoids with controlled stabilizers

Let $p>0$ be a prime and let $M$ be a finite monoid. For each element of $M$, consider the left stabilizer, or dually the right stabilizer, and its minimal ideal. Assume that the maximal subgroup of this minimal ideal is a $p'$-group, meaning that its order is not divisible by $p$. Write $C(\mathbb C M)$ and $C(kM)$ for the Cartan matrices of the corresponding monoid algebras.

Cartan nonsingularity conjecture for controlled stabilizers. If $C(\mathbb C M)$ is nonsingular, then $C(kM)$ is nonsingular for every field $k$ of characteristic $p$.

The conjecture is motivated by results for regular monoids and monoids with aperiodic stabilizers, both of which are presented as evidence for it. Its general case remains open in the source.