Axenovich–Martin conjecture on complete partitions of bipartite-free graphs

Let $H$ be a bipartite graph that is not a forest. Write $f(r,H)$ for the least integer $k$ such that every $H$-free $(r,k)$-graph does not exist, and write $\operatorname{ex}(N,H)$ for the maximum number of edges in an $H$-free graph on $N$ vertices. An $H$-free $(r,k)$-graph is a graph whose vertex set is partitioned into $r$ parts of size at most $k$, with at least one edge between every two parts. Axenovich–Martin conjecture. There is a positive constant $c=c(H)$ such that for any positive integers $r$ and $k$ satisfying

$$\operatorname{ex}(rk,H)>c\binom{r}{2},$$

there is an $H$-free $(r,k)$-graph, equivalently $f(r,H)<k$. If true, this would show that the trivial lower bound gives the correct order of magnitude and remove the logarithmic factor from the known upper bound for $f(r,H)$.