The girth analogue for hereditary graph classes

For a graph $F$ with at least one cycle, let its girth be the length of its shortest cycle. A graph is $F$-free if it contains no subgraph isomorphic to $F$, and an induced subgraph is obtained by restricting a graph to a subset of its vertices. A graph is $b_F$-colourable if its chromatic number is at most $b_F$.

Girth analogue conjecture. For every graph $F$ with at least one cycle, there exists a constant $b_F$ and graphs $G$ of arbitrarily large chromatic number and the same girth as $F$ such that every $F$-free induced subgraph of $G$ is $b_F$-colourable.

This conjecture asks whether the property established in the paper for classes defined by a lower bound on odd girth also holds for ordinary girth. The supplied text gives no resolution, so the conjecture remains open.