Manifold conjecture for Barvinok rank-two matrix complexes

For positive integers $m$ and $n$, let $B_{m,n}$ denote the complex of $m\times n$ matrices of Barvinok rank two. Manifold conjecture. For all $m$ and $n$, the complex $B_{m,n}$ is a manifold. This conjecture is suggested by the derived cross-polytopal decomposition of $B_{3,n}$; no general resolution is given in the source.