  The perfect matching index of a cubic graph $G$, denoted by $\pi(G)$, is the
smallest number of perfect matchings that cover all the edges of $G$. According
to the Berge-Fulkerson conjecture, $\pi(G)\le5$ for every bridgeless cubic
graph~$G$. The class of graphs with $\pi\ge 5$ is of particular interest as
many conjectures and open problems, including the famous cycle double cover
conjecture, can be reduced to it. Although nontrivial examples of such graphs
are very difficult to find, a few infinite families are known, all with
circular flow number $\Phi_c(G)=5$. It has been therefore suggested [Electron.
J. Combin. 23 (2016), $\#$P3.54] that $\pi(G)\ge 5$ might imply $\Phi_c(G)\ge
5$. In this article we dispel these hopes and present a family of cyclically
$4$-edge-connected cubic graphs of girth at least $5$ (snarks) with $\pi\ge 5$
and $\Phi_c\le 4+\frac23$.
