How to contract an essentially 6-connected graph to a 5-connected graph

It is shown that an essentially 6-connected graph G on at least 13 vertices can be contracted to a 5-connected graph H   such that 0<|V(G)|-|V(H)|<50<|V(G)|-|V(H)|<5. The bounds 13 and 5 are sharp, and no such result holds for higher connectivity.