  A graph $H$ is common if the number of monochromatic copies of $H$ in a
2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the
random colouring. We prove that, given $k,r>0$, there exists a $k$-connected
common graph with chromatic number at least $r$. The result is built upon the
recent breakthrough of Kr\'a\v{l}, Volec, and Wei who obtained common graphs
with arbitrarily large chromatic number and answers a question of theirs.
