K-theoretic DT/PT correspondence conjecture for local surfaces

Let $S$ be a smooth surface, let $X=\operatorname{Tot}_S(K_S)$, let $(d,w)\in\mathbb{N}\times\mathbb{Z}$, and let $n=\gcd(d,w)$. The topological K-theoretic Euler characteristic $\chi_K(\mathbb{T}_X(d)_w)$ is defined by

$$\chi_K(\mathbb{T}_X(d)_w):=\dim_{\mathbb{Q}}K_0^{\mathrm{top}}(\mathbb{T}_X(d)_w)_{\mathbb{Q}}-\dim_{\mathbb{Q}}K_1^{\mathrm{top}}(\mathbb{T}_X(d)_w)_{\mathbb{Q}}.$$

K-theoretic computation conjecture.

$$\chi_K(\mathbb{T}_X(d)_w)=\chi_c(\operatorname{Hilb}(S,n)).$$

This proposes a K-theoretic computation of the quasi-BPS categories for local surfaces, generalizing the corresponding computation for $\mathbb{C}^3$. Here $\chi_c$ denotes the compactly supported Euler characteristic, and the conjectural identity remains open in the stated generality.