Degree and unique completability of anti-diagonal bases

Determine the degree of the projection map \pi_\Omega for every base \Omega formed by a family of non-intersecting anti-diagonal paths in the determinantal matroid M(r,[m]\times[n]), and characterize which of these anti-diagonal bases are uniquely completable.

Background

Theorem 4.1 proves that every family of non-intersecting anti-diagonal paths from the specified boundary points is a base of the determinantal matroid. For such a base, the projection map has a zero-dimensional generic fiber, so its degree equals the generic number of rank-r completions.

The paper gives a partial answer: anti-diagonal bases that are ladders are uniquely completable. Examples show that uniquely completable anti-diagonal bases need not all be ladders, leaving the degree and unique-completability classification for the full family unresolved.