Do atoms in internal-energy superpositions obey the equivalence principle?
In plain words
Since energy has mass, an atom in a quantum superposition of two energy levels has no single definite mass. Whether it still falls exactly like a normal atom is a question that classical physics cannot pose.
Precise statement
In the quantum formulation of the equivalence principle (Zych and Brukner, arXiv 1502.00971), the internal mass-energy operator $M = m_0 + H_{\mathrm{int}}/c^2$ may couple to gravity through operators $M_g$ and $M_i$ that differ off-diagonally. Bound the off-diagonal violation parameters for atoms prepared in coherent superpositions of hyperfine or optical clock states at better than $1e-12$, improving on the about $1e-9$ bound of Rosi et al 2017.
What would settle it
An atom interferometer with test atoms in clock-state superpositions measuring the differential acceleration with sub-1e-12 precision.