The quasi-freeness conjecture for graded symmetric-group harmonics

Let $cmathcal{R}_{k,n}$ and $p_{calpha}$ be as in Section 2, with $d(calpha)$ denoting the degree of $p_{calpha}$. Let $cmathcal{H}_{k,n}$ be the graded space of $S_n$-harmonics that is an orthogonal complement to the ideal $cmathcal{I}_{k,n}$ generated by the $p_{calpha}$ with $d(calpha)>0$. Quasi-freeness conjecture. If $m\leq n$ and, for each $calpha$ with $d(calpha)\leq m$, there is a homogeneous $h_{calpha}\incmathcal{H}_{k,n}$ such that $d(calpha)+\deg(h_{calpha})=m$, and if

$$\sum_{d(\alpha)\leq m}h_{\alpha}p_{\alpha}=0,$$

then $h_{calpha}=0$ for all $calpha$ with $d(calpha)\leq m$. This conjecture would give the stated stability formulas for the graded multiplicities and Hilbert series; it is proved in low degrees and has been checked in several small cases, but is not established in general.