$G_2$-instantons on the spinor bundle of the 3-sphere

We classify $G_2$-instantons admitting $SU(2{)}^3$-symmetries, and construct a new family of examples on the spinor bundle of the 3-sphere, equipped with the asymptotically conical, co-homogeneity one $G_2$-metric of Bryant-Salamon. We also show that, outside of the $SU(2{)}^3$-invariant examples, any other $G_2$-instanton on this metric with the same asymptotic behaviour must have obstructed deformations.