Finiteness of the Grassmann distance degree

Determine whether the Grassmann distance (GD) degree is finite for every irreducible model \mathcal{M}\subseteq \mathrm{pGr}(k,n) with Zariski-dense real points, beyond the established finiteness result for models in \mathrm{pGr}(2,n).

Background

The paper defines the GD degree as the number of complex critical points of the Euclidean trace optimization problem when the data point is a generic rank-k orthogonal projection matrix in \mathrm{pGr}(k,n). Unlike the Euclidean distance degree, whose finiteness follows from the dimension of the Euclidean distance correspondence, the GD correspondence can be reducible and may have positive-dimensional generic fibers.

The authors prove in Theorem 3.6 that every model in \mathrm{pGr}(2,n) has finite GD degree. The unresolved question is whether this finiteness extends to arbitrary ranks $k$ and all irreducible models satisfying the paper’s standing hypotheses.