Rank-two validity of Li’s three conjectures

Determine whether Li’s conjectures for Problems 1, 2, and 3 hold under the sole assumption that the matrix G has rank two, without imposing the additional structural constraints considered for the special rank-at-most-two families.

Background

Several structured matrices arising in eigenvalue perturbation theory have rank at most two, and the paper summarizes cases in which Li’s conjectures are known under additional structural assumptions. The authors ask whether those assumptions can be removed while retaining only the condition rank(G)=2. The question concerns all three formulations: the unitary-matrix comparison in Problem 1 and the nonsingular-matrix comparisons in Problems 2 and 3.

References

This raises the question of whether eq:liconj-1, eq:liconj-2, and eq:liconj-0 hold under the sole assumption $\rank(G)=2$, without these additional structural constraints.

eq:liconj-0:

$c_0\cdot\min_{W\ \mathrm{unitary}}\|W\circ G\|_{\UI} \ge\min_{P\ \mathrm{permutation}}\|P\circ G\|_{\UI}. $

— On a Conjecture Related to Eigenvalue Perturbations  (2609.25552 - Zhang et al., 22 Sep 2026) in Section 5, Discussion