Strong Hanani–Tutte theorem on surfaces

Determine whether the Strong Hanani–Tutte theorem holds on every compact orientable surface without boundary, namely, whether an independently even drawing implies that the graph is embeddable on that surface.

Background

The Strong Hanani–Tutte theorem states that if every pair of independent edges crosses an even number of times, then the graph is planar. The analogous statement is known to hold on the projective plane and the torus, but it fails on orientable surfaces of genus at least four.

The unresolved question concerns the remaining surfaces without boundary, including the surfaces whose genus lies between the established positive and negative cases.