The locally dense induced-subgraph conjecture for $P_5$-free graphs

Let $G$ be a finite simple graph. For an induced subgraph $F$ of $G$ and $v\in V(F)$, let $N_F(v)$ be the neighbours of $v$ in $F$. $P_5$ local-density conjecture. There exists $\delta>0$ such that every $P_5$-free graph $G$ has an induced subgraph $F$ satisfying

$$\chi(F)\ge\delta\chi(G)$$

and

$$\chi(N_F(v))\ge\delta\chi(F)$$

for every $v\in V(F)$. This is presented as a weaker, particularly interesting special case of the colourful induced-subgraph conjecture; the source does not report a resolution.