The　generalized　k-connectivity　κk(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.　(2012)　gave　a　lower　bound　for　the　generalized　3-connectivity　of　the　Cartesian　product　graph　GH　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.　©　2018　Elsevier　Inc.