Higher-uniformity monochromatic clique bound for k-partite colorings

Let $g_k(n)$ be the maximum number of edges in an $n$-vertex iterated blowup of a $k$-uniform edge. A $k$-partite coloring of the complete graph $K_n$ is a coloring arising from a partition into $k$ parts as in the source. Higher-uniformity conjecture. For $k\in\{4,5,6\}$, every $k$-partite coloring of $K_n$ has at most $g_k(n)$ monochromatic copies of $K_k$.

The conjecture seeks to extend the corresponding results proved in the paper for lower uniformities to $k=4,5,6$.