It has recently been shown that the abundance of cold neutral gas may follow a similar evolution as the star formation history. This is physically motivated, since stars form out of this component of the neutral gas and if the case, would resolve the long-standing issue that there is a clear disparity between the total abundance of neutral gas and star-forming activity over the history of the Universe. Radio-band 21-cm absorption traces the cold gas and comparison with the Lyman-α absorption, which traces all of the gas, provides a measure of the cold gas fraction, or the spin temperature, Tspin. The recent study has shown that the spin temperature (degenerate with the ratio of the absorber/emitter extent) appears to be anti-correlated with the star formation density, ψ*, with 1/Tspin undergoing a similar steep evolution as ψ* over redshifts of 0 ≲ z ≲ 3, whereas the total neutral hydrogen exhibits little evolution. Above z ∼ 3, where ψ* shows a steep decline with redshift, there are insufficient 21-cm data to determine whether 1/Tspin continues to follow ψ*. Knowing this is paramount in ascertaining whether the cold neutral gas does trace the star formation over the Universe’s history. We explore the feasibility of resolving this with 21-cm observations of the largest contemporary sample of reliable damped Lyman-α absorption systems and conclude that, while today’s largest radio interferometers can reach the required sensitivity at z ≲ 3.5, the Square Kilometre Array is required to probe higher redshifts.