The defect half-index interpretation conjecture

Let $M_3$ be a weakly negative plumbed three-manifold with defects labelled by highest weights $\nu_v\vec\omega$ at nodes $v\in V_W$, and let $\hat Z_b(M_3;W_{\{\nu_v\}};\tau)$ be the proposed defect invariant. Defect half-index interpretation conjecture. The defect $\hat Z$-invariant equals the half-index of the same three-dimensional $\mathcal N=2$ supersymmetric quantum field theory with the corresponding defect operators included. This conjecture supplies the physical interpretation of the proposed defect invariant; the source does not state a mathematical resolution.