  In this paper we study 7D maximally supersymmetric Yang-Mills on a specific
3-Sasakian manifold that is the total space of an $SO(3)$-bundle over
$\mathbb{C}P^2$. The novelty of this example is that the manifold is not a
toric Sasaki-Einstein manifold. The hyperk\"ahler cone of this manifold is a
Swann bundle with hypertoric symmetry and this allows us to calculate the
perturbative part of the partition function of the theory. The result is also
verified by an index calculation. We also discuss a factorisation of this
result and compare it with analogous results for $S^7$.
