  The reconfiguration graph for the $k$-colourings of a graph $G$, denoted
$R_{k}(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two
colourings are joined by an edge if they differ in colour on exactly one
vertex. For any $k$-colourable $P_4$-free graph $G$, Bonamy and Bousquet proved
that $R_{k+1}(G)$ is connected. In this short note, we complete the
classification of the connectedness of $R_{k+1}(G)$ for a $k$-colourable graph
$G$ excluding a fixed path, by constructing a $7$-chromatic $2K_2$-free (and
hence $P_5$-free) graph admitting a frozen $8$-colouring. This settles a
question of the second author.
