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 ε=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.