  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.
