  A multigraph is locally irregular if the degrees of the end-vertices of every
multiedge are distinct. The locally irregular coloring is an edge coloring of a
multigraph $G$ such that every color induces a locally irregular submultigraph
of $G$. A locally irregular colorable multigraph $G$ is any multigraph which
admits a locally irregular coloring. We denote by ${\rm lir}(G)$ the locally
irregular chromatic index of a multigraph $G$, which is the smallest number of
colors required in the locally irregular coloring of the locally irregular
colorable multigraph $G$. In case of graphs the definitions are similar. The
Local Irregularity Conjecture for 2-multigraphs claims that for every connected
graph $G$, which is not isomorphic to $K_2$, multigraph $^2G$ obtained from $G$
by doubling each edge satisfies ${\rm lir}(^2G)\leq 2$. We show this conjecture
for cacti. This class of graphs is important for the Local Irregularity
Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims
that every locally irregular colorable graph $G$ satisfies ${\rm lir}(G)\leq
3$. At the beginning it has been observed that all not locally irregular
colorable graphs are cacti. Recently it has been proved that there is only one
cactus which requires 4 colors for a locally irregular coloring and therefore
the Local Irregularity Conjecture was disproved.
