Chow–Lam recovery for Grassmannian subvarieties

Prove that a subvariety V ⊂ Gr_ℂ(k,n), defined by k(n−r)+1 generic linear forms in the Plücker coordinates, coincides with its Chow–Lam recovery W_V, as conjectured by Pratt–Ranestad (Conjecture 6.6).

Background

Inspired by amplituhedron maps, Chow–Lam recovery asks whether a variety can be reconstructed from linear projections to smaller Grassmannians. The conjecture formulates a precise recovery criterion via the number of defining linear forms. Proving equality V = W_V would generalize classical Chow form reconstruction and solidify recovery methods in positive-geometry contexts.