Actinide ions for testing the spatial $\alpha-$variation hypothesis

Testing the spatial variation of fine-structure constant $\alpha$ indicated in [Webb et al., Phys. Rev. Lett. 107, 191101 (2011)] with terrestrial laboratory atomic measurements requires at least $\dot{\alpha}/\alpha \sim 10^{-19}~\textrm{y}^{-1}$ sensitivity. We conduct a systematic search of atomic systems for such a test that have all features of the best optical clock transitions leading to possibility of the frequency measurements with fractional accuracy on the level of $10^{-18}$ or better and have a factor of 100 extra enhancement of $\alpha$-variation in comparisons to experimental frequency ratio measurement accuracy. We identify the pair of actinide Cf$^{15+}$ and Es$^{16+}$ ions as the best system for a test of spatial $\alpha-$variation hypothesis as it satisfies both of these requirements and have sufficiently simple electronic structure to allow for high-precision predictions of all atomic properties required for rapid experimental progress.