Non-spin extension of the tilted spacetime positive mass theorem

Let $(M^{n\geq 3},g,k,\mathcal{E},\Sigma)$ be an $n$-dimensional complete asymptotically flat initial data set with arbitrary ends and a non-compact boundary, satisfying the interior dominant energy condition $\mu-|J|\geq 0$ and the tilted boundary dominant energy condition

$$H_{\Sigma} \pm \cos\alpha\, \operatorname{tr}_{\Sigma}k \geq \sin\alpha\, |k(\eta, \cdot)^{\top}|$$

for some $\alpha\in [0,\frac{\pi}{2}]$. Let $E_{\mathcal{E}}$ and $P_{\mathcal{E}}$ denote the ADM energy and momentum, and let $(P_{\mathcal{E}})_n$ and $\widehat{P}_{\mathcal{E}}$ be the corresponding normal and tangential components.

Non-spin extension conjecture. Theorem 3.1 holds even if $M$ is not spin; equivalently,

$$E_{\mathcal{E}} \pm \cos\alpha\, (P_{\mathcal{E}})_n \geq \sin\alpha\, |\widehat{P}_{\mathcal{E}}|.$$

The result is proved in the paper for spin initial data sets, whereas the non-spin case is explicitly stated to remain unknown. A dimension-three proof is noted separately, but the general non-spin extension remains open.