Leader–Letzter–Narayanan–Walters extremal density conjecture for product-free sets

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 the odd-occurrence set $\mathcal{O}_\Gamma$ consist of the words in which the total number of occurrences of letters from $\Gamma$ is odd.

Leader–Letzter–Narayanan–Walters conjecture. If $S\subset\mathcal{F}$ is product-free and $d^{\ast}(S)=1/2$, then

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

for some nonempty subset $\Gamma\subset\mathcal{A}$.

The upper Banach density bound $d^{\ast}(S)\leqslant 1/2$ is known, and odd-occurrence sets attain density $1/2$; the conjecture asserts that these are the only extremal examples.