Locally compatible quantum d-to-1 pseudo-telepathy conjecture

Fix $d\geq 2$. A quantum assignment to a label-cover instance is $k$-compatible when the measurements associated with variables in every subinstance of Gaifman-graph diameter at most $k$ are simultaneously diagonalisable. Locally compatible quantum d-to-1 conjecture. Fix $d\geq 2$. For each $\epsilon>0$ and each $k\in\mathbb{N}$ there exists a $d$-to-$1$ instance $\Phi$ (on some alphabet size $n$) such that $\Phi$ admits a perfect $k$-compatible quantum assignment but $\operatorname{sat}(\Phi)<\epsilon$. This is proposed as a pseudo-telepathy analogue of the $d$-to-$1$ Conjecture: quantum players should win perfectly while classical players have arbitrarily small value. The supplied text gives no evidence that the conjecture has been resolved.