$G_2$-instantons on the ALC members of the $\mathbb{B}_7$ family

Using co-homogeneity one symmetries, we construct a two-parameter family of non-abelian $G_2$-instantons on every member of the asymptotically locally conical $\mathbb{B}_7$-family of $G_2$-metrics on $S^3 \times \mathbb{R}^4 $, and classify the resulting solutions. These solutions can be described as perturbations of a one-parameter family of abelian instantons, arising from the Killing vector-field generating the asymptotic circle fibre. Generically, these perturbations decay exponentially to the model, but we find a one-parameter family of instantons with polynomial decay. Moreover, we relate the two-parameter family to a lift of an explicit two-parameter family of anti-self-dual instantons on Taub-NUT $\mathbb{R}^4$, fibred over $S^3$ in an adiabatic limit.