Finite-dimensionality conjecture for the non-commuting Harmonic system

Let $\mathbb{C}\langle X_n\rangle$ be the free associative algebra on $X_n$, and let $Har_n$ denote the non-commuting Harmonic system defined earlier in the paper. Write $Har_n^{(d)}$ for its homogeneous component of degree $d$.

Finite-dimensionality conjecture. The non-commuting Harmonic system $Har_n$ is finite dimensional for every $n$.

The paper gives computed Hilbert-series data for $n\leq 5$, including dimensions $1,2,9,946,\ldots$ for the first listed values. The supplied text does not state a proof or resolution, so the assertion remains open here.