Hardness conjecture for approximate 3-coloring

Let $G$ be a $3$-colorable graph on $n$ nodes. Hardness of approximate $3$-coloring. For some fixed $\gamma > 0$, there is no polynomial time algorithm that, given $G$, returns a valid $n^\gamma$ coloring of $G$. This conjecture is used as a hardness assumption for lower bounds on bicriteria masked low-rank approximation; its resolution status is not specified in the source.