  Transit surveys indicate that there is a deficit of Neptune-sized planets on
close-in orbits. If this ``Neptune desert' is entirely cleared out by
atmospheric mass loss, then planets at its upper edge should only be marginally
stable against photoevaporation, exhibiting strong outflow signatures in
tracers like the metastable helium triplet. We test this hypothesis by carrying
out a 12-night photometric survey of the metastable helium feature with
Palomar/WIRC, targeting seven gas-giant planets orbiting K-type host stars.
Eight nights of data are analyzed here for the first time along with reanalyses
of four previously-published datasets. We strongly detect helium absorption
signals for WASP-69b, HAT-P-18b, and HAT-P-26b; tentatively detect signals for
WASP-52b and NGTS-5b; and do not detect signals for WASP-177b and WASP-80b. We
interpret these measured excess absorption signals using grids of Parker wind
models to derive mass-loss rates, which are in good agreement with predictions
from the hydrodynamical outflow code ATES for all planets except WASP-52b and
WASP-80b, where our data suggest that the outflows are much smaller than
predicted. Excluding these two planets, the outflows for the rest of the sample
are consistent with a mean energy-limited outflow efficiency of $\varepsilon =
0.41^{+0.16}_{-0.13}$. Even when we make the relatively conservative assumption
that gas-giant planets experience energy-limited outflows at this efficiency
for their entire lives, photoevaporation would still be too inefficient to
carve the upper boundary of the Neptune desert. We conclude that this feature
of the exoplanet population is a pristine tracer of giant planet formation and
migration mechanisms.
