A proof of the interlacing conjecture for the singular values of Sylvester-Kac matrices

Boros and Rózsa (Linear Algebra Appl. 421 (2007) 407--416) reduced the singular value problem for the odd-order Sylvester--Kac matrix $P\in\mathbb{R}^{(2m+1)\times(2m+1)}$ to those for $Q^{(m)}\in\mathbb{R}^{(m+1)\times m}$ and $S^{(m)}\in\mathbb{R}^{m\times(m+1)}$ via a permutation similarity trans