Nonsingular rank-one version of Problem 3

Determine whether the nonsingular-matrix version of Problem 3 holds for rank-one matrices G and arbitrary unitarily invariant norms, namely whether there exists a dimension-independent constant c_2 such that \[ c_2\min_{W\ \mathrm{nonsingular}}\|W\circ G\|_{\mathrm{UI}}\,\|W^{-1}\circ G^{\mathsf T}\|_{\mathrm{UI}}\geq \min_{P\ \mathrm{permutation}}\|P\circ G\|_{\mathrm{UI}}^2. \]

Background

The paper explains that Romeo and Tilli proved the corresponding statements for rank-one matrices when the minimization is restricted to unitary matrices. The authors note that the analogous statement with the minimizing matrix allowed to be any nonsingular matrix remains unresolved. This is a structured special case of Li’s third conjecture, which compares minimization over nonsingular matrices with minimization over permutation matrices for a unitarily invariant norm.