Properness conjecture for moment maps of complete toric Kähler–Ricci solitons

Let $(M,g,J,\bm T^n)$ be a complete toric Kähler–Ricci soliton, with moment map

$$\bm\mu\colon M\to\mathfrak t^*.$$

Properness conjecture. The moment map $\bm\mu$ is proper.

Properness is an important condition in the theory of non-compact toric Kähler manifolds because it permits the use of Duistermaat–Heckman theory and helps define weighted analogues of energy functionals. The conjecture is motivated by the need to understand complete toric Kähler–Ricci solitons beyond the free-action case; the cited source does not provide a resolution.