Rectangular-flat-torus grid contact representation conjecture

A bipartite graph has a vertex partition $V=W\cup B$ with no edges within $W$ or within $B$. A grid contact representation assigns the vertices in $W$ to horizontal segments and the vertices in $B$ to vertical segments, with segments disjoint except that a segment of one kind may touch both ends of a segment of the other kind at interior points, and such a contact occurs only for adjacent vertices. A toroidal graph is a graph that can be embedded on the torus, and a rectangular flat torus is a flat torus obtained by identifying opposite sides of a rectangle.

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

Grid contact representations are known for toroidal bipartite graphs on a flat torus that need not be rectangular. This conjecture asks whether the absence of loops allows the torus to be chosen rectangular.