  The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$, which was
introduced by Chartrand et al.(1984) is a generalization of the concept of
vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently,
Li et al. gave a lower bound for the generalized $3$-connectivity of the
Cartesian product graph $G \square H$ and proposed a conjecture for the case
that $H$ is $3$-connected. In this paper, we give two different forms of lower
bounds for the generalized $3$-connectivity of Cartesian product graphs. The
first lower bound is stronger than theirs, and the second confirms their
conjecture.
