Gutt–Hutchings consistency conjecture for capacities of star-shaped domains

Let $U$ be a bounded star-shaped domain, and let $c_k^{\text{EH}}(U)$, $c_k^{\text{GH}}(U)$, and $c_k(U)$ denote the Ekeland–Hofer, Gutt–Hutchings, and capacities considered in the paper, respectively. For every $k\in \mathbb{Z}_{\geq 1}$, Gutt–Hutchings' conjecture.

$$c_k^{\text{EH}}(U)=c_k^{\text{GH}}(U)=c_k(U).$$

The equality of the first two capacities is conjectured by Gutt and Hutchings for bounded star-shaped domains. The case $k=1$, namely equality with the minimal action, is known for convex bodies, while the consistency for all $k$ remains open in the stated generality.