Does the proton-to-electron mass ratio vary in time or space?
In plain words
Molecular vibrations depend on the ratio of proton to electron mass, while atomic transitions mostly do not, so comparing them tests whether this ratio changes. Both laboratory molecular clocks and ancient molecular absorption in distant galaxies are used.
Precise statement
Determine whether $\mu = m_p/m_e$ varies, either as $d\ln(\mu)/dt$ today from comparisons of molecular (for example HD+, H2+, N2+, $\mathrm{Sr_2}$) and atomic clocks, or as $\Delta \mu/\mu$ at redshift $z > 1$ from $\mathrm{H_2}$ and $\mathrm{CH_3OH}$ absorption, at the $10^{-17}$ per year and $10^{-7}$ levels respectively.
What would settle it
A molecular-clock to atomic-clock ratio tracked for several years with fractional uncertainty below $10^{-17}$, or a radio methanol absorption measurement with sub-$10^{-7}$ precision.
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