  Conformally equivariant quantization is a peculiar map between symbols of
real weight $\delta$ and differential operators acting on tensor densities,
whose real weights are designed by $\lambda$ and $\lambda+\delta$. The
existence and uniqueness of such a map has been proved by Duval, Lecomte and
Ovsienko for a generic weight $\delta$. Later, Silhan has determined the
critical values of $\delta$ for which unique existence is lost, and conjectured
that for those values of $\delta$ existence is lost for a generic weight
$\lambda$. We fully determine the cases of existence and uniqueness of the
conformally equivariant quantization in terms of the values of $\delta$ and
$\lambda$. Namely, (i) unique existence is lost if and only if there is a
nontrivial conformally invariant differential operator on the space of symbols
of weight $\delta$, and (ii) in that case the conformally equivariant
quantization exists only for a finite number of $\lambda$, corresponding to
nontrivial conformally invariant differential operators on $\lambda$-densities.
The assertion (i) is proved in the more general context of IFFT (or AHS)
equivariant quantization.
