The graded hat-harmonics conjecture

Let $\widehat{D}_k=\sum_{i=1}^n\partial_i^{k+1}x_i$ for $k\geq1$, and define the space of hat-harmonics by

$$\widehat{\mathcal{H}}_{\mathbf{x}}=\{f(\mathbf{x})\in\mathbb{R}[\mathbf{x}]\mid \widehat{D}_k f(\mathbf{x})=0,\ \forall k\geq1\}.$$

The symmetric group $\mathfrak S_n$ acts by permuting variables, and the space is graded. The graded hat-harmonics conjecture. As a graded $\mathfrak S_n$-module, $\widehat{\mathcal{H}}_{\mathbf{x}}$ is isomorphic to the space of $\mathfrak S_n$-harmonics. This would imply, in particular, the experimentally observed dimension $n!$.