The asymptotic Ramsey-function conjecture for mean Ramsey–Turán numbers

Let $H$ be a graph and let $k$ be a positive integer. The mean Ramsey–Turán number $RT(n,H,k\text{-mean})$ is the maximum number of edges in a $k$-mean colored graph with $n$ vertices that contains no monochromatic copy of $H$. Let $r(H,k\text{-mean})$ be the smallest integer $r$ for which there exists a complete $r$-partite graph such that every $k$-mean edge-coloring of it contains a monochromatic copy of $H$.

Mean Ramsey-function conjecture.

$$RT(n,H,k\text{-mean})=\left(1-\frac{1}{r(H,k\text{-mean})-1}\right){n \choose 2}+o(n^2).$$

For ordinary and local Ramsey–Turán numbers, an analogous asymptotic relation follows from the Erdős–Stone theorem, but no such relation was known for mean Ramsey–Turán numbers. The conjecture asserts that the same asymptotic description holds in the mean setting.