Cyclotomic Catalan triangle conjecture

For a positive integer $\mu$, let $\omega=e^{2i\pi/\mu}$ and define the cyclotomic analogue by

$$C_{n,k}^{(\mu)}=C_{n,k}(\omega,\ldots,\omega).$$

Assume that $\mu$ divides $n+1$.

Cyclotomic Catalan conjecture.

$$C_{n,n}^{(\mu)}=\frac{1}{\mu}\binom{n+1}{(n+1)/\mu}.$$

These specializations are presented as cyclotomic analogues related to the cyclic sieving phenomenon. The excerpt provides no proof or resolution status.