Polynomial-time branchwidth conjecture for graphs embeddable in the torus and projective plane

Let $H$ be a graph that can be embedded both in the torus and in the projective plane. Consider graphs excluding $H$ as a minor.

Polynomial-time branchwidth conjecture. Branchwidth can be computed in polynomial time on $H$-minor-free graphs.

Exact polynomial-time computation of branchwidth is known for graph classes excluding a single-crossing graph as a minor, while the corresponding question for graphs excluding a graph embeddable in both the torus and the projective plane remains open.