Extremal 3-product-free set conjecture for free semigroups

Let $\mathcal{A}$ be a finite set and let $\mathcal{F}$ be the free semigroup with alphabet $\mathcal{A}$. If $S\subset\mathcal{F}$ is 3-product-free and $d^{\ast}(S)=1/3$, consider the two types of label sets

$$\{w\in\mathcal{F}:\text{the sum of the labels of letters in }w\text{ is }1\bmod 3\},$$

and

$$\{w\in\mathcal{F}:\text{the sum of the labels of letters in }w\text{ is }1,2\bmod 6\}.$$

Extremal 3-product-free set conjecture. One of the following holds: either the letters of $\mathcal{A}$ can be labelled by $\mathbb{Z}/3\mathbb{Z}$ so that $S$ is contained in the first set, or the letters can be labelled by $\mathbb{Z}/6\mathbb{Z}$ so that $S$ is contained in the second set.

This is the free-semigroup analogue of the classification of extremal 3-sum-free sets in the non-negative integers. The density bound is established, but the extremal structure for $k=3$ is explicitly left open.