The Chebyshev interlacing inequality conjecture

Let $\kappa\geq2$ be even and fix $1\leq j\leq\lfloor\kappa/2\rfloor$. Let $T_\kappa$ be the Chebyshev polynomial used in the paper, and choose real numbers satisfying

$$\cos\left(\frac{(j+1)\pi}{\kappa}\right)<a<\cos\left(\frac{j\pi}{\kappa}\right)<b<\cos\left(\frac{(j-1)\pi}{\kappa}\right),$$

with $T_\kappa(a)=T_\kappa(b)$. Chebyshev interlacing conjecture. Then

$$\cos\left(\frac{j\pi}{\kappa}\right)-a>b-\cos\left(\frac{j\pi}{\kappa}\right).$$

The paper gives no proof or resolution.