Kedlaya-set optimality conjecture for product-free subsets

Let $G$ be a compact group equipped with its Haar probability measure, and let a Kedlaya set be a set of the form

$$K_{x,B}=\{g\in G: gx\in B\ \text{and}\ g(B)\subseteq X\setminus B\},$$

where $G$ acts on a set $X$, $B\subseteq X$, and $x\in X$. A measurable subset of $G$ is product-free if $gh\notin A$ for all $g,h\in A$. Kedlaya-set optimality conjecture. There exists an absolute constant $C>0$ such that $G$ has a Kedlaya set $K$ for which every measurable product-free subset of $G$ has Haar measure at most $C\mu(K)$. This proposes a general optimality principle for the known construction of large product-free sets; whether it holds for every compact group is left open in the source.