A Penrose-like lower bound for ADM momentum

Let $(M,g,K) fin \mathcal{M}_{\textnormal{AF}}$ and let $\{\Sigma_r\}_{r\in[r_0,\infty)}\subset\mathcal{M}$, where

$$\Sigma_r=\{r\}\times\mathbb{S}^2.$$

Suppose that the inner boundary $\Sigma_{r_0}$ is a generalized apparent horizon and that the family $\{\Sigma_r\}_{r\in(r_0,\infty)}$ satisfies the strictly untrapped condition. The ADM momentum conjecture. One has

$$|\mathbf{P}|\leq\int_{r_0}^{\infty}\int_{\mathbb{S}^2}r^2\bigl(|\mathbf{J}|_g-N^{-1}|\mathbf{J}_0|\bigr)\,dA_{\sigma(r)}\,dr.$$

This conjecture would extend the momentum estimate underlying the Penrose inequality beyond spherical symmetry and provide a refined lower bound for the ADM mass in settings where the ADM momentum need not vanish.