Kim, Kim and Liu's conjecture for the Ramsey–Turán function $\rho(3,6,\delta)$

Let $\rho(3,6,\delta)$ denote the asymptotic maximum edge density of an $n$-vertex graph with independence number at most $\delta n$ admitting a 2-edge-coloring with no monochromatic blue $K_3$ and no monochromatic red $K_6$. Kim, Kim and Liu's conjecture. For any sufficiently small $\delta>0$,

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

The paper proves the upper bound with coefficient $2.1025$ in place of $2$, leaving a small gap from the matching lower-bound construction; hence the conjecture remains open.