Global equivariant Hilbert-function conjecture for Haiman neighborhoods

Let $\mathscr{P}_r$ be the set of $r$-dimensional partitions, including the empty partition. For $\lambda\in\mathscr{P}_r$, let $A_{\lambda}$ be the coordinate ring of the Haiman neighborhood and let $H(A_{\lambda};\theta_1,\ldots,\theta_r)$ be its equivariant Hilbert function. Global equivariant Hilbert-function conjecture. For $r\geq2$,

$$\sum_{\lambda\in\mathscr{P}_r}\left(Q^{|\lambda|}H(A_{\lambda};\theta_1,\ldots,\theta_r)\prod_{\mathbf{i}=(i_1,\ldots,i_r)\in\lambda}(1-u\theta_1^{i_1}\cdots\theta_r^{i_r})(1-v\theta_1^{-i_1}\cdots\theta_r^{-i_r})\right) =\exp\left(\sum_{n=1}^{\infty}\frac{(1-u^n)(1-v^n)Q^n}{n(1-\theta_1^n)\cdots(1-\theta_r^n)}\right).$$

This is a local-to-global generating-series prediction for equivariant Hilbert functions. The paper proves finite-order cases in several settings, including smooth proper toric threefolds and, modulo $Q^8$, all smooth projective threefolds, but the full identity remains open.