Rainbow-free equinumerous 4-colorings of Z24k+8 and Z24k+16

Determine whether there exists a rainbow-free equinumerous 4-coloring of Z_{24k+r} for natural number k and r∈{8,16}.

Background

The paper discusses a rainbow-free equinumerous 4-coloring of Z_{24k} originating from the construction of Conlon, Jungic, and Radoicic. Motivated by that construction, the authors explicitly ask whether analogous rainbow-free equinumerous 4-colorings exist for the two residue classes Z_{24k+8} and Z_{24k+16}.