Coclosed $G_2$-structures on $\text{SU}(2)^2$-invariant cohomogeneity one manifolds

We consider two different $\text{SU}(2)^2$-invariant cohomogeneity one manifolds, one non-compact $M=\mathbb{R}^4 \times S^3$ and one compact $M=S^4 \times S^3$, and study the existence of coclosed $\text{SU}(2)^2$-invariant $G_2$-structures constructed from half-flat $\text{SU}(3)$-structures. For $\mathbb{R}^4 \times S^3$, we prove the existence of a family of coclosed (but not necessarily torsion-free) $G_2$-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed $G_2$-structure constructed from a half-flat $\text{SU}(3)$-structure is in this family. For $S^4 \times S^3$, we prove that there are no $\text{SU}(2)^2$-invariant coclosed $G_2$-structures constructed from half-flat $\text{SU}(3)$-structures.