Criterion for properness in general locally compact groups

Determine an explicit and effective criterion for two subsets L and H of an arbitrary locally compact group G to satisfy the properness relation L pitchfork H, modulo the equivalence relation ~ between subsets of G.

Background

The paper introduces binary relations pitchfork and sim on the power set of a locally compact group G to recast properness of actions as an internal property of G rather than of the homogeneous space G/H. For reductive Lie groups, a complete criterion is given via Cartan projections (Theorem 3.15).

However, beyond the reductive setting, the general problem of characterizing when L pitchfork H holds (modulo sim) has not been resolved. The authors emphasize that by "criterion" they mean a practical, effective test rather than a purely formal restatement.