Gap-preserving compression for commuting-operator strategies

Let $\mathscr{G}=(G_n)_{n\in\mathbb{N}}$ be a sequence of nonlocal games, and let the complexity of a sequence refer to the complexity of its $n$th game. Gap-preserving compression conjecture. There exists a computable map $\texttt{GappedCompress}_{co}$ that, given $\mathscr{G}$, outputs a sequence $\mathscr{G}'=(G_n')_{n\in\mathbb{N}}$ of complexity $O(\log n)$ such that, whenever the complexity of $\mathscr{G}$ is at most $\mathrm{poly}(n)$, for every $n\in\mathbb{N}$,

$$\omega_{co}(G_n)=1\Longrightarrow\omega_{co}(G_n')=1,$$

and

$$\omega_{co}(G_n)\leq\frac{1}{2}\Longrightarrow\omega_{co}(G_n')\leq\frac{1}{2}.$$

The conjecture would provide a gapped compression procedure for commuting-operator strategies and could establish the hardness of approximating the commuting-operator value. Its status is unknown in the source.