Looseness width of 5-connected triangulations on the torus

Let G be a triangulation on a closed surface and c:V(G)→{1,2,3,…,3+k} a color assignment of the vertices of G. Then a face uvw of G is said to be heterochromatic for c if its three corners u, v and w receive three distinct colors. Furthermore, G is said to be k-loosely tight if there is a heterochromatic face of G for any surjection c:V(G)→{1,2,3,…,3+k}. The looseness of G, denoted by ξ(G), is defined as the minimum k such that G is k-loosely tight. We show that if G is 5-connected triangulation on the torus, then ξ(G) is independent of the embedding of G.