The $\widehat Z$-invariant as a Rozansky–Witten invariant

Let $M_3$ be a 3-manifold, let $G$ be the relevant group, and let

$$X={\mathcal M}_H(G,D^2)=\text{“}T^*\operatorname{Gr}_{G_{\mathbb C}}\text{”}$$

be the moduli space associated with the disk $D^2$, viewed conceptually as the cotangent bundle of the affine Grassmannian. Denote by $\widehat Z(M_3)$ the $q$-series invariant and by $Z_{\mathrm{RW}[X]}(M_3)$ the Rozansky–Witten invariant with target $X$. The $\widehat Z$–Rozansky–Witten conjecture.

$$\widehat Z(M_3)=Z_{\mathrm{RW}[X]}(M_3).$$

The proposal is motivated by the interpretation of $\widehat Z$-invariants as Rozansky–Witten invariants with the non-compact, infinite-dimensional target ${\mathcal M}_H(G,D^2)$; the paper provides evidence, but a general proof is not supplied.