  We show that the $E_1$-equivalence $C^\bullet(S^2) \simeq H^\bullet(S^2)$
does not intertwine the inclusion of constant loops into the free loop space
$S^2 \to LS^2$. That is, the isomorphism $HH_\bullet(H^\bullet(S^2)) \cong
H^\bullet(LS^2)$ does not preserve the obvious maps to $H^\bullet(S^2)$ that
exist on both sides. We give an explicit computation of the defect in terms of
the $E_\infty$-structure on $C^\bullet(S^2)$. Finally, we relate our
calculation to recent work of Poirier-Tradler on the string topology of $S^2$.
