The half-data conjecture for motivic Hilbert-scheme polynomials

Let $\mathsf P_d(t)$ denote the polynomial in the motivic formula for Hilbert schemes of $d$ points, and suppose that $\mathsf P_e(t)$ is known for every integer $e$ with $1\leq e<d$. Half-data conjecture. For $d>3$, the number of new data required to compute $\mathsf P_d(t)$ is at most

$$\left\lceil\frac{d-1}{2}\right\rceil.$$

Here the new data are the motives $[\operatorname{Hilb}^d({\mathbb A}^i)]$ for $i=1,\ldots,\left\lceil(d-1)/2\right\rceil$. This is presented as a motivic refinement of the corresponding bound for higher-dimensional partition numbers.