Upper Banach density conjecture for product-free subsets of free groups

Let $\mathscr{A}$ be a finite alphabet, and let $\mathbf{F}_{\mathscr{A}}$ be the free group on $\mathscr{A}$. A subset of a group is product-free if it contains no product of two of its elements, and its upper Banach density is computed with respect to the natural weighting on irreducible words described in the paper.

Upper Banach density conjecture. No product-free subset $S\subset\mathbf{F}_{\mathscr{A}}$ has upper Banach density exceeding

$$\frac12.$$

This is the proposed free-group analogue of the paper's upper-density bound for product-free subsets of free semigroups. The conjecture remains open in the source.