On the Hardness of $4$-to-$1$ Games with Perfect Completeness

We prove that for all $\varepsilon>0$, there exists a positive integer $k$ such that given a $4$-to-$1$ game $Ψ$ with alphabet size at most $k$, it is $\mathbf{NP}$-hard to distinguish between the case that $\mathrm{val}(Ψ)=1$ and the case that $\mathrm{val}(Ψ)\leq \varepsilon$. This confirms the $4