Partition of triples into independent subhypergraphs and 4-cliques

Construct or prove the existence of a partition of the triples of an m-element set into roughly m/2 independent 3-uniform subhypergraphs and roughly m^3/8 4-cliques.

Background

The paper proposes a construction for S_4(P_3∪K_2)-free 4-uniform hypergraphs that would improve the known lower bound if a particular partition existed. The required partition simultaneously decomposes the triples of an m-element set into approximately m/2 independent subhypergraphs and approximately m^3/8 4-cliques. The authors explicitly state that the existence of such a partition is unresolved.