  In 2014, Gromov vaguely conjectured that a sequence of manifolds with
nonnegative scalar curvature should have a subsequence which converges in some
weak sense to a limit space with some generalized notion of nonnegative scalar
curvature. The conjecture has been made precise at an IAS Emerging Topics
meeting: requiring that the sequence be three dimensional with uniform upper
bounds on diameter and volume, and a positive uniform lower bound on MinA,
which is the minimum area of a closed minimal surface in the manifold. Here we
present a sequence of warped product manifolds with warped circles over
standard spheres, that have circular fibres over the poles whose length
diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We
prove this sequence converges in the $W^{1,p}$ sense for $p<2$ to an extreme
limit space that has nonnegative scalar curvature in the distributional sense
as defined by Lee-LeFloch and that the total distributional scalar curvature
converges. This paper only requires expertise in smooth Riemannian Geometry,
smooth minimal surfaces, and Sobolev Spaces. In a second paper, requiring
expertise in metric geometry, the first two authors prove intrinsic flat and
Gromov-Hausdorff convergence of our sequence to this extreme limit space and
investigate its geometric properties.
