Properness conjecture for automorphism groups of strictly pseudoconvex CR manifolds

Let $M$ be a connected, strictly pseudoconvex CR manifold, and let $\scr A(M)$ denote its CR automorphism group. An action of a Lie group on a space is proper when the map

$$G\times X\to X\times X,\qquad (g,x)\mapsto(g\cdot x,x)$$

is proper.

Properness conjecture. The action of $\scr A(M)$ on $M$ should be proper unless $M$ is CR equivalent to the sphere or to the Heisenberg group with its standard CR structure.

This is proposed by analogy with the corresponding theorem for conformal manifolds. In the compact case, properness implies compactness of $\scr A(M)$, so the claim includes the conjecture that the automorphism group is compact for compact strictly pseudoconvex 3-manifolds other than the sphere; its general status is not specified in the source.