Kim–Kim–Liu's local Ramsey–Turán density conjecture for $(K_3,K_6)$

For $\delta>0$, let $\rho(3,6,\delta)$ 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 $K_6$. Kim–Kim–Liu's conjecture. For sufficiently small $\delta>0$,

$$\rho(3,6,\delta)=\frac{5}{12}+\frac{\delta}{2}+2\delta^2.$$

The source describes the right-hand side as attained by a construction, so the conjecture concerns the matching upper bound for sufficiently small $\delta$.