Converse direction for absolute weights and Cauchy completeness in general proarrow equipments

Determine whether, for an arbitrary proarrow equipment ι: K → M, every weight W whose weighted colimits are absolute necessarily admits a right adjoint V, and equivalently, ascertain whether an object A in K is Cauchy complete if and only if A has all absolute weighted colimits. This is known to hold for the enriched-category equipment V-Cat → V-Prof, but it is unknown in general.

Background

In Appendix A.4 the paper reviews absolute weighted (co)limits and Cauchy completeness in the abstract framework of proarrow equipments. Proposition A.4.1 shows that weights given by left adjoints yield absolute colimits. In the classical enriched setting (V-Cat → V-Prof), Street (1983) proved the converse: weights for which all colimits are absolute must be right adjoint weights, and hence Cauchy completeness is equivalent to the existence of all absolute weighted colimits.

The authors point out that it is unknown whether this converse extends from the enriched setting to arbitrary proarrow equipments, leaving a gap in the general theory connecting absolute weights and adjointness, and the corresponding characterization of Cauchy completeness.