Maximal abelianness conjecture for frame representations of commutative semigroups

Let $\mathcal{S}$ be a unital commutative semigroup, possibly with the cancellation property, and let $\pi$ be a frame representation of $\mathcal{S}$. Let $\mathcal{A}$ be the $wot$-closed subalgebra of $B(H)$ generated by $\pi(\mathcal{S})$. Maximal abelianness conjecture. The algebra $\mathcal{A}$ is maximal abelian. This would imply, in particular, that every frame representation of such a semigroup is central; the conjecture is presented as an extension of the preceding result for commutative semigroups whose left-regular-generated algebra is maximal abelian.