An independent third determination of $\alpha$ at 0.1 ppb
In plain words
With two top measurements disagreeing, a third method that shares none of their possible errors could tell which one is wrong. Candidates use other atoms, such as strontium or ytterbium, or the fine splitting of helium energy levels.
Precise statement
Determine $\alpha$ with relative uncertainty below 1e-10 using a method whose systematics are independent of Raman or Bragg recoil interferometry in alkali atoms, for example single-photon large-momentum-transfer interferometry in Sr or Yb, or helium $2^{3}P$ fine structure, which requires the $\alpha^{8}m$ QED corrections (theory is complete at $\alpha^{7}m$ since 2010) and resolution of the disagreement among helium $2^{3}P$ measurements. The answer is a value of $\alpha^{-1}$ with uncertainty below 1.4e-8.
What would settle it
Publication of an $\alpha$ value at below $0.1\,\mathrm{ppb}$ from a non-alkali or non-recoil method.