The joint Ramsey–Turán density conjecture for $(K_3,K_7)$

For $\delta>0$, let $\rho(3,7,\delta)$ be the limiting maximum edge density of an $n$-vertex graph with independence number at most $\delta n$ that admits a red/blue coloring with no red $K_3$ and no blue $K_7$. The $(K_3,K_7)$ density conjecture. For sufficiently small $\delta>0$,

$$\rho(3,7,\delta)=\frac{7}{16}+\frac{\delta}{2}.$$

The source gives a construction proving the corresponding lower bound. The conjecture is the assertion that this construction is asymptotically optimal for sufficiently small $\delta$.