Stability conjecture for near-extremal product-free sets in free semigroups

Let $\mathcal{A}$ be a finite set and let $\mathcal{F}$ be the free semigroup with alphabet $\mathcal{A}$. For a nonempty subset $\Gamma\subset\mathcal{A}$, let $\mathcal{O}_\Gamma$ be the odd-occurrence set consisting of words with an odd total number of occurrences of letters from $\Gamma$.

Stability conjecture. For each $\delta>0$, there exists $\varepsilon>0$ such that, whenever $S\subset\mathcal{F}$ is product-free and

$$d^{\ast}(S)>\frac12-\varepsilon,$$

there is an odd-occurrence set $\mathcal{O}_\Gamma$ such that

$$d^{\ast}(S\setminus\mathcal{O}_\Gamma)<\delta.$$

The conjecture asks whether product-free sets whose density is close to the extremal value $1/2$ must be close in upper Banach density to an odd-occurrence set. The source presents this as an open stability problem; exact extremal structure is known in the relevant free-semigroup results, but this quantitative strengthening is not.