The Erdős–Ginzburg–Ziv bound for finite commutative semigroups

Let $\mathcal{S}$ be a finite commutative semigroup. Write $\mathsf D(\mathcal{S})$ for its large Davenport constant, let $\mathsf E(\mathcal{S})$ be the invariant defined by the existence of a subsequence with sum-preserving deletion of $\kappa(\mathcal{S})$ terms, and set

$$\kappa(\mathcal{S})=\left\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})}\right\rceil\exp(\mathcal{S}).$$

Erdős–Ginzburg–Ziv conjecture for semigroups. For any finite commutative semigroup $\mathcal{S}$,

$$\mathsf E(\mathcal{S})\leq \mathsf D(\mathcal{S})+\kappa(\mathcal{S})-1.$$

For finite abelian groups this generalizes the classical equality $\mathsf E(G)=\mathsf D(G)+|G|-1$. The reverse inequality holds trivially when $\mathcal{S}$ has an identity element, so the conjecture would imply equality for finite commutative monoids; the source gives no resolution of the general semigroup case.