  In this paper, we observe new phenomena related to the structure of
3-manifolds satisfying lower scalar curvature bounds. We construct
warped-product manifolds of almost nonnegative scalar curvature that converge
to pulled string spaces in the Sormani-Wenger intrinsic flat topology. These
examples extend the results of Lee-Naber-Neumayer \cite{LNN} to the case of
dimension $3$. As a consequence, we produce the first counterexample to a
conjecture of Sormani \cite{SormaniConj} on the stability of the Geroch
Conjecture. Our example tests the appropriate hypothesis for a related
conjecture of Gromov. On the other hand, we demonstrate a $W^{1,p}$-stability
statement ($1\leq p<2$) for the Geroch Conjecture in the class of warped
products.
