Rectangular-flat-torus tessellation representation conjecture

A toroidal graph is a graph that can be embedded on the torus, and a tessellation representation of a graph is a grid contact representation of the bipartite graph whose vertices are the faces and vertices of the graph and whose edges are their incidences. A rectangular flat torus is a flat torus obtained by identifying opposite sides of a rectangle.

Rectangular-flat-torus tessellation conjecture. Every toroidal graph without loops has a tessellation representation on the rectangular flat torus.

Mohar and Rosenstiehl constructed tessellation representations of toroidal graphs on flat tori, but their construction may require a non-rectangular flat torus because of graphs with loops. The conjecture asks whether excluding loops suffices to obtain a representation on a rectangular flat torus.