A 2-planar graph is a graph that can be drawn in the plane so that each edge is crossed at most twice. It is known that every 2-planar graph satisfies that $|E(G)|\le 5(|V(G)|-2)$. This implies that every 2-planar graph contains a vertex of degree at most 9 and thus every 2-planar is 10-colorable. On the other hand, $K_7$ is a 2-planar graph. So my question is whether there is a 2-planar graph with chromatic number 8 or 9 or 10? Thanks for pointers!

Note the $K_8$ is not 2-planar!