  Interfacing quantum mechanics and gravity is one of the great open questions
in natural science. Micromechanical oscillators have been suggested as a
plausible platform to carry out these experiments. We present an experimental
design aiming at these goals, inspired by Schm\"ole et al., Class. Quantum
Grav. 33, 125031 (2016). Gold spheres weighing on the order a milligram will be
positioned on large silicon nitride membranes, which are spaced at
submillimeter distances from each other. These mass-loaded membranes are
mechanical oscillators that vibrate at $\sim 2$ kHz frequencies in a drum mode.
They are operated and measured by coupling to microwave cavities. First, we
show that it is possible to measure the gravitational force between the
oscillators at deep cryogenic temperatures, where thermal mechanical noise is
strongly suppressed. We investigate the measurement of gravity when the
positions of the gravitating masses exhibit significant quantum fluctuations,
including preparation of the massive oscillators in the ground state, or in a
squeezed state. We also present a plausible scheme to realize an experiment
where the two oscillators are prepared in a two-mode squeezed motional quantum
state that exhibits nonlocal quantum correlations and gravity the same time.
Although the gravity is classical, the experiment will pave the way for testing
true quantum gravity in related experimental arrangements. In a
proof-of-principle experiment, we operate a 1.7 mm diameter Si$_3$N$_4$
membrane loaded by a 1.3 mg gold sphere. At 10 mK temperature, we observe the
drum mode with a quality factor above half a million at 1.7 kHz, showing strong
promise for the experiments. Following implementation of vibration isolation,
cryogenic positioning, and phase noise filtering, we foresee that realizing the
experiments is in reach by combining known pieces of current technology.
