Nonnegative-coefficient conjecture for nested Hilbert-scheme Euler characteristics

Let $n\geq 1$ and $m\geq 1$. Let $\operatorname{Hilb}^{n,n-1}(\mathbb{A}^2)$ be the nested Hilbert scheme, let $\eta:\operatorname{Hilb}^{n,n-1}(\mathbb{A}^2)\to\operatorname{Hilb}^{n}(\mathbb{A}^2)$ send $(I_1,I_2)$ to $I_1$, and let $M(m)=\mathcal{O}(m)\otimes\mathcal{O}_{Z_n}$, where $Z_n$ is the zero fiber of $\operatorname{Hilb}^{n}(\mathbb{A}^2)$. Define

$$\mathcal{N}^{(m)}_n(q,t)=\chi_{\operatorname{Hilb}^{n,n-1}(\mathbb{A}^2),\eta^*M(m)}(q,t).$$

Nonnegative-coefficient conjecture. For every $m\geq 1$, $\mathcal{N}^{(m)}_n(q,t)$ is a polynomial in $q$ and $t$ with nonnegative integer coefficients. The authors further conjecture, for $m=1$, that

$$\mathcal{N}^{(1)}_n(q,t)=N_n(q,t),$$

where $N_n(q,t)$ is the combinatorial nested $q,t$-Catalan series.

These conjectures seek a positive combinatorial interpretation of Euler characteristics obtained from the Atiyah–Bott–Lefschetz formula on nested Hilbert schemes. The source presents them as computer-experiment-based conjectures; no resolution is supplied.