On the Colin de Verdière numbers of Cartesian graph products

Let G be a connected graph with Colin de Verdière number μ(G). We study the behaviour of μ with respect to the Cartesian product of graphs. We conjecture that if G=G₁□G₂, with G₁,G₂ connected, then μ(G)?μ(G₁)+μ(G₂) and prove that μ(G)?μ(G₁)+h(G₂)-1, where h is the Hadwiger number (i.e. the order of the largest clique minor). In addition we provide an explicit construction of a Colin de Verdière matrix with corank μ(G₁)+μ(K_n) for the graph G=G₁□K_n.