$3$-Colouring $P_t$-free graphs without short odd cycles

For any odd $t\ge 9$, we present a polynomial-time algorithm that solves the $3$-colouring problem, and finds a $3$-colouring if one exists, in $P_{t}$-free graphs of odd girth at least $t-2$. In particular, our algorithm works for $(P_9, C_3, C_5)$-free graphs, thus making progress towards determining the complexity of $3$-colouring in $P_t$-free graphs, which is open for $t\ge 8$.