It's well-known that the eigenvalues of the Clement-Kac-Sylvester tridiagonal matrix

$$\begin{pmatrix}
0 & n-1 & 0 & \dots & 0 \\\ 
1 & 0 & n-2 & \dots & 0\\\
0 & 2 & 0 & \dots & 0 \\\ 
\vdots & \vdots & \vdots & \ddots & 1 \\\ 
0 & 0 & 0 & n-1 & 0
\end{pmatrix}$$

are $\pm (n), \pm (n-2), ..., (\pm1 \; or \; 0)$. Please refer to <https://mathoverflow.net/questions/132186/regarding-a-paper-by-paul-a-clement-on-tridiagonal-matrices/132261#132261> and <https://math.stackexchange.com/questions/405670/regarding-a-paper-by-paul-a-clement-on-tridiagonal-matrices> for details.

My question is do we know the eigenvalues of the leading principal submatrix of the Clement-Kac-Sylvester tridiagonal matrix $(1<r\le n/2)$:

$$\begin{pmatrix}
0 & n-1 & 0 & ... & 0 \\\ 
1 & 0 & n-2 &  & ... \\\
0 & 2 & 0 & ... & 0 \\\ 
... &  & ... &  & n-r+1 \\\ 
0 & ... & 0 & r-1 & 0
\end{pmatrix}$$

or do we have any lower/upper bound on the eigenvalues of this submatrix.