  Manolescu and Piccirillo recently initiated a program to construct an exotic
$S^4$ or $\# n \mathbb{CP}^2$ by using zero surgery homeomorphisms and
Rasmussen's $s$-invariant. They find five knots that if any were slice, one
could construct an exotic $S^4$ and disprove the Smooth $4$-dimensional
Poincar\'e conjecture. We rule out this exciting possibility and show that
these knots are not slice. To do this, we use a zero surgery homeomorphism to
relate slice properties of two knots \textit{stably} after a connected sum with
some $4$-manifold. Furthermore, we show that our techniques will extend to the
entire infinite family of zero surgery homeomorphisms constructed by Manolescu
and Piccirillo. However, our methods do not completely rule out the possibility
of constructing an exotic $S^4$ or $\# n \mathbb{CP}^2$ as Manolescu and
Piccirillo proposed. We explain the limits of these methods hoping this will
inform and invite new attempts to construct an exotic $S^4$ or $\# n
\mathbb{CP}^2$. We also show a family of homotopy spheres constructed by
Manolescu and Piccirillo using annulus twists of a ribbon knot are all
standard.
