Subpolynomial growth conjecture for even-edge cliques

Let $K_p$ be the complete graph on $p$ vertices, and let $f(n,H)$ be the smallest number of colors in an edge coloring of $K_n$ in which every copy of $H$ intersects at least one color class in an odd number of edges. Subpolynomial clique-growth conjecture. For any positive integer $p\geq 4$ with $p\equiv 0,1\pmod{4}$, one has

$$f(n,K_p)=n^{o(1)}.$$

The conjecture extends the stated general upper bound, whose polynomial exponent tends to zero as $p$ increases, and concerns the variant of the Erdős–Gyárfás problem for cliques with an even number of edges. The supplied source does not indicate whether it has been resolved.