Are there any experiments which have probed how such "negative" energies gravitate?
There is The Archimedes Experiment
Archimedes is an INFN-funded pathfinder experiment aimed at verifying the feasibility of measuring the interaction of vacuum fluctuations with gravity. The final experiment will measure
the force exerted by the gravitational field on a Casimir cavity whose vacuum energy is modulated with a superconductive transition, by using a balance as a small force detector.
Archimedes is two-year project devoted to test the most critical experimental aspects, in particular the balance resonance frequency and quality factor, the thermal modulation efficiency
and the superconductive sample realization.