  We address the uniqueness of the minimal couplings between higher-spin fields
and gravity. These couplings are cubic vertices built from gauge non-invariant
connections that induce non-abelian deformations of the gauge algebra. We show
that Fradkin-Vasiliev's cubic 2-s-s vertex, which contains up to 2s-2
derivatives dressed by a cosmological constant $\Lambda$, has a limit where:
{(i)} $\Lambda\to 0$; {(ii)} the spin-2 Weyl tensor scales
{\emph{non-uniformly}} with s; and {(iii)} all lower-derivative couplings are
scaled away. For s=3 the limit yields the unique non-abelian spin 2-3-3 vertex
found recently by two of the authors, thereby proving the \emph{uniqueness} of
the corresponding FV vertex. We extend the analysis to s=4 and a class of spin
1-s-s vertices. The non-universality of the flat limit high-lightens not only
the problematic aspects of higher-spin interactions with $\Lambda=0$ but also
the strongly coupled nature of the derivative expansion of the fully nonlinear
higher-spin field equations with $\L\neq 0$, wherein the standard minimal
couplings mediated via the Lorentz connection are \emph{subleading} at energy
scales $\sqrt{|\Lambda|}<< E<< M_{\rm p}$. Finally, combining our results with
those obtained by Metsaev, we give the complete list of \emph{all} the
manifestly covariant cubic couplings of the form 1-s-s and 2-s-s, in Minkowski
background.
