Primary binomial ideals in positive characteristic are mesoprimary

Let $\Bbbk$ be an algebraically closed field of arbitrary characteristic, let $Q$ be a finitely generated commutative monoid, and let $\Bbbk[Q]$ be its monoid algebra. Mesoprimary binomial ideal conjecture. Every binomial primary ideal in $\Bbbk[Q]$ is mesoprimary. This would clarify the combinatorics of primary binomial ideals in positive characteristic and potentially show that the paper's construction of binomial primary decompositions is stronger than necessary. The statement is presented as an unresolved missing point.