Necessity conjecture for Hilbert-series deviation of q-deformed quasi-invariants

Let $m$ and $n$ be positive integers, let $q$ be a nonzero complex number, and let $Q_{m,q}(n)$ and $Q_m(n,\mathbb{C})$ denote the $q$-deformed and ordinary $m$-quasi-invariant polynomial algebras, respectively. Necessity conjecture. If the Hilbert series of $Q_{m,q}(n)$ differs from that of $Q_m(n,\mathbb{C})$, then $q$ is a primitive $p$-th root of unity for an integer $p$ satisfying

$$\frac{mn(n-2)+{n\choose{2}}}{{n\choose{2}}-1}\leq p\leq mn.$$

The claim asserts that the root-of-unity condition from Theorem 5.6 is not only sufficient but necessary; no resolution is supplied in the source, so the necessity remains open.