The moment formula conjecture for powers of Voiculescu's circular operator

Let $T$ be Voiculescu's circular operator, let $\tau$ denote the trace, and let $k,n\in{\mathbf N}$. Moment formula conjecture.

$$\tau\big(((T^*)^kT^k)^n\big)=\frac{n^{nk}}{(nk+1)!}.$$

The formula is proved for $k=1$, and it is also elementary for $n=1$ and provable for $n=2$; further cases were checked computationally. The general assertion remains open in the source.