The exact-degree conjecture for the motivic polynomial of punctual Hilbert schemes

For each positive integer $d$, let $\mathsf P_d(t)$ be the polynomial appearing in the motivic formula for Hilbert schemes of $d$ points on smooth varieties. Exact-degree conjecture. For every $d>3$, the polynomial $\mathsf P_d(t)$ has degree exactly $d-2$.

The main theorem only gives the upper bound $\deg \mathsf P_d\leq d-2$; the conjecture asserts that the top coefficient never vanishes.