  The Petersen colouring conjecture states that every bridgeless cubic graph
admits an edge-colouring with $5$ colours such that for every edge $e$, the set
of colours assigned to the edges adjacent to $e$ has cardinality either $2$ or
$4$, but not $3$. We prove that every bridgeless cubic graph $G$ admits an
edge-colouring with $4$ colours such that at most $\frac45\cdot|V(G)|$ edges do
not satisfy the above condition. This bound is tight and the Petersen graph is
the only connected graph for which the bound cannot be decreased. We obtain
such a $4$-edge-colouring by using a carefully chosen subset of edges of a
perfect matching, and the analysis relies on a simple discharging procedure
with essentially no reductions and very few rules.
