The spectral interval conjecture for positive semi-simple TERP data

Fix $u_1,\ldots,u_n\in{\mathds C}$, $\xi\in S^1$ with $\Re((u_i-u_j)/\xi)<0$ for $i<j$, and $T\in M(n\times n,{\mathds R})$ upper triangular with $T_{ii}=1$ and $T+T^{tr}$ positive definite. Let the corresponding mixed TERP-structure have weight $w$. Spectral interval conjecture. The mixed TERP-structure is pure and polarized, and its spectral numbers at infinity lie in

$$\left(\frac{w-1}{2},\frac{w+1}{2}\right).$$

This strengthens the preceding positivity criterion by adding a predicted spectral bound; the source gives no proof or resolution.