  The rigidity theorems of Llarull and Marques-Neves, which show two different
ways scalar curvature can characterize the sphere, have associated stability
conjectures. Here we produce the first examples related to these stability
conjectures. The first set of examples demonstrates the necessity of including
a condition on the minimum area of all minimal surfaces to prevent bubbling
along the sequence. The second set of examples constructs sequences that do not
converge in the Gromov-Hausdorff sense but do converge in the volume preserving
intrinsic flat sense. In order to construct such sequences, we improve the
Gromov-Lawson tunnel construction so that one can attach wells and tunnels to a
manifold with scalar curvature bounded below and only decrease the scalar
curvature by an arbitrarily small amount. Moreover, we are able to generalize
both the sewing construction of Basilio, Dodziuk, and Sormani, and the
construction due to Basilio, Kazaras, and Sormani of an intrinsic flat limit
with no geodesics.
