Hilbert-series conjecture for the differential closure of the central zonotopal ideal

Let ${\mathcal{A}}$ be a rank $r$ central multiarrangement in $\mathbb{C}^n$ with $|{\mathcal{A}}|=m$. Define

$$I'_{{\mathcal{A}}}=\left( \lambda_L^{\rho_{{\mathcal{A}}}(L)}, d\lambda_L^{\rho_{{\mathcal{A}}}(L)}: L \subseteq \mathbb{C}^n \text{ a line} \right) \subseteq \Omega_n.$$

Here $\Omega_n$ is the superspace algebra, $\lambda_L$ is the linear form associated with the line $L$, and $\rho_{{\mathcal{A}}}(L)$ is the corresponding rank-nullity exponent. Let $T_{{\mathcal{A}}}(x,y)$ denote the Tutte polynomial of ${\mathcal{A}}$. Hilbert-series conjecture. The bigraded Hilbert series of the quotient $\Omega_n/I'_{{\mathcal{A}}}$ satisfies

$${\mathrm{Hilb}}\left(\Omega_n/I'_{{\mathcal{A}}};q,t\right)=(1+t)^rq^{m-r}T_{{\mathcal{A}}}\left(\frac{1}{1+t},\frac{1}{q}\right).$$

This conjecture would give a Tutte-polynomial specialization for the differential closure that simultaneously reflects the classical central and internal zonotopal algebras. The relevant Hilbert series is known, but a basis for the corresponding Macaulay inverse is not known, and the internal zonotopal algebra remains comparatively difficult to understand.