A class number problem for imaginary cyclic number fields of 2-power degrees

In 2024, M. K. Ram proved that the class number of an imaginary cyclic quartic number field is never equal to a prime $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$. Here we greatly generalize this result to the case of the non-quadratic imaginary cyclic number fields of $2$-power degrees and not necessarily prime class numbers.