The complete-graph conjecture for 2-mean Ramsey–Turán numbers

Let $K_m$ be the complete graph on $m$ vertices. Write $RT(n,K_m,2\text{-mean})$ for the maximum number of edges in a 2-mean colored graph with $n$ vertices containing no monochromatic copy of $K_m$. Let $RT(n,K_m,2)$ be the ordinary 2-color Ramsey–Turán number, let $R(K_m,2)$ be the two-color Ramsey number, and let $t(n,k)$ denote the number of edges in the Turán graph $T(n,k)$.

Complete-graph 2-mean conjecture.

$$RT(n,K_m,2\text{-mean})=RT(n,K_m,2)=t(n,R(K_m,2)-1).$$

The claim strengthens the preceding asymptotic conjecture in the case of complete graphs and two colors: it predicts exact equality with the ordinary Ramsey–Turán number and the corresponding Turán number.