Parity conjecture for Quot schemes of points on smooth threefolds

Let r,d\begin{in}\mathbb{N}\end{in} be two positive integers and let $X$ be an irreducible smooth threefold. For a point $[\mathcal F]$ of the Quot scheme $\operatorname{Quot}_r^d X$, write $\mathsf T_{[\mathcal F]}\operatorname{Quot}_r^d X$ for its tangent space. Parity conjecture. For every $[\mathcal F]\in \operatorname{Quot}_r^d X$, one has

$$\dim_{\mathbb{C}}\mathsf T_{[\mathcal F]}\operatorname{Quot}_r^d X\equiv r\cdot d\pmod 2.$$

This generalizes parity results for monomial subschemes and homogeneous finite-length modules. The conjecture has been disproved by the authors, who provide a ten-dimensional family of zero-dimensional schemes in a Hilbert scheme of points.