Geometric inequality conjecture for asymptotically Euclidean immersions

Let $n$ and $q$ be integers such that $n\geq3$ and $0<q<n/2$, and let $\psi\colon M^n\to\mathbb{R}^d$, with $d>n$, be an asymptotically Euclidean immersion of order $\tau>\tau_q$. Let $S_{2q}$ and $S_{2q+1}$ denote the corresponding higher mean curvatures, let $\bar Z$ be the Euclidean position vector, and assume that $S_{2q}$ is integrable and nonnegative.

Geometric inequality conjecture. One has

$$(n-2q)\int_M S_{2q}\,\mathrm{d}M+(2q+1)\int_M\langle S_{2q+1},\bar Z\rangle\,\mathrm{d}M\geq0,$$

with equality if and only if $(M,\psi^*\bar\delta)$ is isometric to Euclidean space.

Together with the conjectured existence of asymptotically Euclidean isometric immersions, this would imply the positive mass conjecture for the Gauss--Bonnet--Chern mass through the paper's mass identity. The source gives no resolution of this conjecture.