  In 1985, Erd\H{o}s and Ne\v{s}et\v{r}il conjectured that the square of the
line graph of a graph $G$, that is $L(G)^2$, can be colored with
$\frac{5}{4}\Delta(G)^2$ colors. This conjecture implies the weaker conjecture
that the clique number of such a graph, that is $\omega(L(G)^2)$, is at most
$\frac{5}{4}\Delta(G)^2$. In 2015, \'Sleszy\'nska-Nowak proved that
$\omega(L(G)^2)\le \frac{3}{2}\Delta(G)^2$. In this paper, we prove that
$\omega(L(G)^2)\le \frac{4}{3}\Delta(G)^2$. This theorem follows from our
stronger result that $\omega(L(G)^2)\le \frac{\sigma(G)^2}{3}$ where $\sigma(G)
:= \max_{uv\in E(G)} d(u) + d(v)$, is the Ore-degree of the graph $G$.
