Quadratic motivic Donaldson–Thomas conjecture for real threefolds

Let $X$ be a smooth projective threefold over $\mathbb{R}$, equipped with an isomorphism

$$K_X\cong L^{\otimes 2}$$

for some line bundle $L$ on $X$. For $n\geq 0$, let $\tilde{I}_n$ be the quadratic degree of the motivic virtual fundamental class associated to $\operatorname{Hilb}^n(X)$. Let $\operatorname{ob}_2(X)$ denote the obstruction sheaf on $\operatorname{Hilb}^2(X)$, let $e(\operatorname{ob}_2(X))$ be its Euler class, and let $\widetilde{\deg}$ denote quadratic degree. Quadratic motivic Donaldson–Thomas conjecture. One has

$$\sum_{n\geq 0}\tilde{I}_nq^n=M(-q^2)^{-\widetilde{\deg}(e(\operatorname{ob}_2(X)))}.$$

The conjecture is motivated by the computed values for $\mathbb{P}^3_{\mathbb{R}}$, where the first nonzero invariants agree with the expansion of $M(-q^2)^{-10}$. The source does not provide evidence resolving the conjecture, and its stated range includes all even $n$, where the localization method encounters positive-dimensional fixed loci.