Hilbert-series conjecture for the nonnegative spherical Hall algebra

Let $\mathcal{H}^{\mathrm{sph},\geq 0}_{\nu}$ be the subspace of the spherical Hall algebra component $\mathcal{H}^{\mathrm{sph}}_{\nu}$ consisting of functions supported on $\mathbf{Coh}^{\geq 0}_{\nu}$, and let $A^{\geq 0}_{g,r,d}(0)$ denote the corresponding constant-term invariants. Write $z^{\nu}$ for the monomial associated with $\nu\in(\mathbb{N}^2)$ and let $\operatorname{Exp}$ denote the plethystic exponential. Hilbert-series conjecture. In the ring $\mathbb{N}[[z^{(0,1)},z^{(1,0)}]]$,

$$\sum_{\nu}\text{dim}\left(\mathcal{H}^{\mathrm{sph},\geq 0}_{\nu}\right)z^{\nu}=\operatorname{Exp}\left(\sum_{\nu}A^{\geq 0}_{g,r,d}(0)z^{\nu}\right).$$

The nonnegative spherical Hall algebra has finite-dimensional graded components, making this a natural analogue of the Kac denominator or Hilbert-series formula. The source gives no resolution.