Uniqueness conjecture for maximal-density product-free sets in free semigroups

Let $\mathscr{A}$ be a finite set, and let $\mathcal{F}_{\mathscr{A}}$ be the free semigroup on $\mathscr{A}$. A subset $S\subset\mathcal{F}_{\mathscr{A}}$ is product-free if it contains no product of two of its elements, and $d^*(S)$ denotes its upper asymptotic density. For a nonempty subset $\Gamma\subset\mathscr{A}$, let $\mathcal{O}_{\Gamma}$ be the corresponding odd-occurrence set.

Uniqueness conjecture. If $S\subset\mathcal{F}_{\mathscr{A}}$ is product-free and

$$d^*(S)=\frac12,$$

then there is a nonempty subset $\Gamma\subset\mathscr{A}$ such that

$$S\subset\mathcal{O}_{\Gamma}.$$

The conjecture asserts that the odd-occurrence sets are the only extremal constructions of product-free sets of maximal density, up to containment. The paper notes that several non-isomorphic extremal constructions arise from this family, but does not establish uniqueness.