Exact SUSY Wilson loops on $S^3$ from $q$-Virasoro constraints

Using the ideas from the BPS/CFT correspondence, we give an explicit recursive formula for computing supersymmetric Wilson loop averages in 3d $\mathcal{N}=2$ Yang-Mills-Chern-Simons $U(N)$ theory on the squashed sphere $S_b^3$ with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level ${\kappa }_2$ and Fayet-Illiopoulos parameter ${\kappa }_1$. For these values of ${\kappa }_1$ and ${\kappa }_2$ the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent $D^2\times S^1$ theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.