Steinberg's non-singularity conjecture for Cartan matrices of monoid algebras

Let $M$ be a finite monoid, and let $k$ be a field. Write $kM$ for the monoid algebra of $M$ over $k$, and assume that the Cartan matrix of the monoid algebra $\mathbb C M$ is non-singular. Steinberg's conjecture. The Cartan matrix of $kM$ is non-singular for every field $k$. This conjecture proposes that the non-singularity of Cartan matrices for finite group algebras partially extends to finite monoid algebras; the supplied source identifies it as a conjecture but gives no resolution status.