  Say a graph $G$ is a {\em pentagraph} if every cycle has length at least
five, and every induced cycle of odd length has length five. N. Robertson
proposed the conjecture that the Petersen graph is the only pentagraph that is
three-connected and internally 4-connected, but this was disproved by M.
Plummer and X. Zha in 2014. Plummer and Zha conjectured that every 3-connected,
internally 4-connected pentagraph is three-colourable. We prove this: indeed,
we will prove that every pentagraph is three-colourable.
