The colourful induced-subgraph conjecture for forests

Let $G$ be a finite simple graph. For a vertex $v\in V(G)$, write $N_G[v]=N_G(v)\cup\{v\}$, and call $G$ $\varepsilon$-colourful if

$$\chi(G\setminus N_G[v])<\varepsilon\chi(G)$$

for every $v\in V(G)$. For an induced subgraph $F$ of $G$, use the same definition with $F$ in place of $G$. Colourful induced-subgraph conjecture. For every $\varepsilon>0$ and every forest $T$, there exists $\delta>0$ such that every $T$-free graph $G$ has an $\varepsilon$-colourful induced subgraph $F$ with $\chi(F)\ge\delta\chi(G)$. This asserts the existence of a locally dense induced subgraph carrying a linear proportion of the chromatic number. It is known for $T=P_4$ and for stars, but the source says it is undecided for $T=P_5$ and even for the two-edge matching.