Quantum n-coloring is undecidable for every n $\ge$ 3

The quantum $n$-coloring problem, given graph $G$, asks whether there exists a perfect quantum strategy for the $n$-coloring game of $G$. Previously it was known that quantum $n$-coloring is undecidable only for $n=3$. We extend this to all $n\ge 3$ with one elementary reduction. Letting $G^{\mathrm{\prime }}=(G\mathrm{\square }{K}_3)\vee K_{n-3}$, we show that $G$ is quantum $3$-colorable if and only if $G^{\mathrm{\prime }}$ is quantum $n$-colorable.