Conjecture 4.2: Complete intersection equations for Jordan type loci for arbitrary stable partitions

Establish that for any stable partition Q of length l and any partition P with generic commuting Jordan type Q (i.e., ?(P) = Q), the closure of the locus W_P in the nilpotent commutator N_{J_Q} is an irreducible complete intersection of codimension l(P) − l, defined by equations of degree at most l.

Background

The authors prove in Theorem 1.3 that for stable two-part partitions Q = (u, u − r), the closures of the loci W_{Q,k,l} in N_{J_Q} are irreducible complete intersections with explicit equations, leveraging the Box Theorem and tropical rank computations.

Conjecture 4.2 generalizes this result to stable partitions Q of arbitrary length l. The paper notes that the conjecture is straightforward for l = 1 and established for l = 2 by the main theorem here, but remains unproven in general (particularly for l > 3). The authors suggest that an inductive approach and a more detailed analysis of the outer layer of the Box D^{-1}(Q) may yield a proof, but such developments lie beyond the scope of this work.