Schmeichel's basis-number conjecture for toroidal graphs

A graph $G$ is toroidal if it can be embedded on the torus, and its basis number $\operatorname{bn}(G)$ is the smallest integer $k$ for which $G$ has a cycle-space basis in which every edge belongs to at most $k$ basis elements. Schmeichel's conjecture. Every toroidal graph has basis number $3$, that is,

$$\operatorname{bn}(G)=3.$$

This conjecture extends MacLane's characterization of planar graphs, for which the basis number is $2$, to graphs embeddable on the torus. Schmeichel proved the general upper bound $\operatorname{bn}(G)\leq 2g+2$ for graphs of genus $g$, but the stated toroidal case is not resolved in the supplied source.