Bull-free graphs and $χ$-boundedness

A bull is a graph obtained from a four-vertex path by adding a vertex adjacent to the two middle vertices of the path. A graph $G$ is bull-free if no induced subgraph of $G$ is a bull. We prove that for all $k,t\in \mathbb N$, if $G$ is a bull-free graph of clique number at most $k$ and every triang