  A graph is locally irregular if the degrees of the end-vertices of every edge
are distinct. An edge coloring of a graph G is locally irregular if every color
induces a locally irregular subgraph of G. A colorable graph G is any graph
which admits a locally irregular edge coloring. The locally irregular chromatic
index X'irr(G) of a colorable graph G is the smallest number of colors required
by a locally irregular edge coloring of G. The Local Irregularity Conjecture
claims that all colorable graphs require at most 3 colors for a locally
irregular edge coloring. Recently, it has been observed that the conjecture
does not hold for the bow-tie graph B, since B is colorable and requires at
least 4 colors for a locally irregular edge coloring. Since B is a cactus graph
and all non-colorable graphs are also cacti, this seems to be a relevant class
of graphs for the Local Irregularity Conjecture. In this paper we establish
that X'irr(G)<= 4 for all colorable cactus graphs.
