  We study the $SO(3)$ sector of the $\mathcal{N}=1^*$ mass deformation of
$\mathcal{N}=4$ super Yang-Mills on $S^4$. The gravity dual of this sector is
$\mathcal{N}=2$ supergravity coupled to two hypermultiplets. The scalar fields
in the hypermultiplets span a quaternionic-Kahler manifold that is described by
the coset $G_{2,2}/SU(2)\times SU(2)$.
  We use the $\mathcal{N}=2$ supergravity dual to study field configurations in
the bulk that feature analytical solutions, and compute the corresponding $S^4$
free energy using the procedure of holographic renormalization. We find that
the free energy of these configurations is quadratic in the mass and show that
it is devoid of unphysical ambiguities, hence providing an analytical
prediction for the $\mathcal{N}=1$ four-sphere partition function at large 't
Hooft coupling in the planar limit.
