Lipsman's conjecture on properness and the CI property for nilpotent homogeneous spaces

Let $G$) be a connected and simply connected nilpotent Lie group, and let $L$ and $H$ be connected closed subgroups. Consider the natural action of $L$ on $G/H$.

Lipsman's conjecture. The following two conditions are equivalent: (i) the $L$-action on $G/H$ is proper; (ii) the $L$-action on $G/H$ has the CI property.

Lipsman raised this question as an extension of the equivalence known for triples of real reductive Lie groups. Its status is not resolved in the supplied text.