  Llarull's Theorem states that any Riemannian metric on the $n$-sphere which
has scalar curv{\-}ature greater than or equal to $n(n-1)$, and whose distance
function is bounded below by the unit sphere's, is isometric to the unit
sphere. Gromov later posed the {\emph{Spherical Stability Problem}}, which
probes the flexibility of this fact. We give a resolution to this problem in
dimension $3$. Informally, the main result asserts that a sequence of
Riemannian $3$-spheres whose distance functions are bounded below by the unit
sphere's with uniformly bounded Cheeger isoperimetric constant and scalar
curvatures tending to $6$ must approach the round $3$-sphere in the volume
preserving Sormani-Wenger Intrinsic Flat sense. The argument is based on a
proof of Llarull's Theorem due to Hirsch-Kazaras-Khuri-Zhang using spacetime
harmonic functions.
