Is $\alpha$ drifting today at the $10^{-19}$ per year level?
In plain words
Two clocks whose ticking depends differently on $\alpha$ would slowly drift apart if $\alpha$ changed. So far no drift has been seen at about one part in $10^{18}$ per year.
Precise statement
Present laboratory bounds from optical clock ratios (for example the Yb+ E3/E2 comparison) give $\left|\mathrm{d}\,\mathrm{ln}(\alpha)/\mathrm{d}t\right|$ of order $10^{-18}$ per year. Determine whether $\mathrm{d}\,\mathrm{ln}(\alpha)/\mathrm{d}t$ is nonzero at the $10^{-19}$ per year level using multi-year ratios of clocks with large differential sensitivity, including the Th-229 nuclear transition.
What would settle it
A multi-year ratio of a Th-229 nuclear clock (or highly charged ion clock) to an optical atomic clock with fractional uncertainty below $10^{-18}$, analyzed for a linear drift.
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