The fixed-point formula conjecture for the motivic generating series of points on affine three-space

Let $X$ be the Hilbert scheme of points on ${\mathbb A}_{\kappa}^3$, with the torus ${{\mathbb G}_{\mathrm m}}$-action whose isolated fixed points correspond to monomial ideals. Let $P$ range over the 3-dimensional plane partitions corresponding to these fixed points, and let $(T_PX)_+$ and $(T_PX)_-$ denote the positive- and negative-weight subspaces of the tangent space at $P$. Define

$$Z_{{\mathbb A}_{\kappa}^3}(T)=\sum_{n=0}^{\infty}[\operatorname{Hilb}^n({\mathbb A}_{\kappa}^3)]^{\mathrm{virt}}T^n.$$

Fixed-point formula conjecture. The motivic generating series satisfies

$$Z_{{\mathbb A}_{\kappa}^3}(T)=\sum_{n=0}^{\infty}\left({\mathbb L}^{-\frac{1}{2}(\dim(T_PX)_+-\dim(T_PX)_-)}\right)T^n.$$

This conjecture proposes a fixed-point expression for the motivic generating function of the Hilbert schemes of points on affine three-space, despite the torus action not satisfying the hypotheses of the cited localization result. The fixed points are indexed by monomial ideals, equivalently by 3-dimensional plane partitions; the source provides no resolution of the conjecture.