The general Ramsey–Turán formula conjecture for $\rho(3,q)$

For each integer $q\ge 5$, let $\rho(3,m)$ denote the Ramsey–Turán density for graphs admitting a 2-edge-coloring with no monochromatic blue $K_3$ and no monochromatic red $K_m$, and let $r(3,q)$ denote the classical Ramsey number. The general Ramsey–Turán formula conjecture.

$$\left\{ \begin{array}{ll} \rho(3,2q-1)=\frac{1}{2}\left(1-\frac{1}{r(3,q)-1}\right), &\\ \rho(3,2q)=\frac{1}{2}\left(1-\frac{1}{r(3,q)}\right). & \end{array} \right.$$

This statement appears in an ignored concluding passage and is presented as a conjecture about the wide-open values of $\rho(3,q)$ for $q\ge 8$; its resolution is not supplied, so it remains open.