  The UV finiteness found in calculations of the 4-point amplitude in $\mathcal
N=5$ supergravity at loop order $L=3, 4$ has not been explained, which
motivates our study of the relevant superspace invariants and on-shell
superamplitudes for both $\mathcal N=5$ and $\mathcal N=6$. The local 4-point
superinvariants for $L = 3,4$ are expected to have nonlinear completions whose
6-point amplitudes have non-vanishing SSL's (soft scalar limits), violating the
behavior required of Goldstone bosons. For $\mathcal N=5$, we find at $L=3$
that local 6-point superinvariant and superamplitudes, which might cancel these
SSL's, do not exist. This rules out the candidate 4-point counterterm and thus
gives a plausible explanation of the observed $L=3$ finiteness. However, at $L=
4$ we construct a local 6-point superinvariant with non-vanishing SSL's, so the
SSL argument does not explain the observed $L=4$ $\mathcal N=5$ UV finiteness.
For $\mathcal N=6$ supergravity there are no 6-point invariants at either $L=
3$ or 4, so the SSL argument predicts UV finiteness.
