Erdős et al.'s Ramsey–Turán density conjecture for odd cliques

Let $r(3,q)$ denote the two-color Ramsey number for a triangle and a clique of order $q$. For $q\geq 2$, the Ramsey–Turán density $\rho(3,2q-1)$ is defined using graphs admitting a red/blue coloring with no red $K_3$ and no blue $K_{2q-1}$, together with vanishing independence ratio. Erdős et al.'s conjecture.

$$\rho(3,2q-1)=\frac{1}{2}\left(1-\frac{1}{r(3,q)-1}\right).$$

This conjecture is part of the program of determining $\rho(3,q)$; the values for sufficiently large clique orders remain open.