Rainbow-free equinumerous 4-colorings of Z8k

Determine whether there exists a rainbow-free equinumerous 4-coloring of the cyclic group Z_{8k} for any natural number k>1.

Background

The paper proves that every equinumerous 4-coloring of Z_8 contains a rainbow 4-term arithmetic progression. It then asks whether this phenomenon persists for larger cyclic groups whose orders are multiples of 8, specifically whether a rainbow-free equinumerous 4-coloring exists for Z_{8k} when k is not 1.