  Gromov and Sormani conjectured that a sequence of three dimensional
Riemannian manifolds with nonnegative scalar curvature and some additional
uniform geometric bounds should have a subsequence which converges in some
sense to a limit space with generalized notion of nonnegative scalar curvature.
In this paper, we study the pre-compactness of a sequence of three dimensional
warped product manifolds with warped circles over standard $\mathbb{S}^2$ that
have nonnegative scalar curvature, a uniform upper bound on the volume, and a
positive uniform lower bound on the MinA, which is the minimum area of closed
minimal surfaces in the manifold. We prove that such a sequence has a
subsequence converging to a $W^{1, p}$ Riemannian metric for all $p<2$, and
that the limit metric has nonnegative scalar curvature in the distributional
sense as defined by Lee-LeFloch.
