Horowitz-Myers AdS soliton conjecture

A ``new" positive energy conjecture was conjectured back in 1999 by Horowitz and Myers [arXiv:hep-th/9808079], which somehow escaped my attention till Steve McCormick mentioned it to me in a tweet (X-post? sounds weird though). I will briefly outline what it is, but the technicalities will be left to the reader as an exercise. The idea is essentially to take the near-extremal p-brane solution and double Wick rotate it, so as to obtain the AdS soliton with periodicity. The interest then would be to work with metrics that look like 

\[g_{\mu \nu }=\tilde{g}_{\mu \nu }+h_{\mu \nu }\]

and fall-off conditions w.r.t $r$, where $\tilde{g}_{\mu \nu }$ is the AdS soliton metric. Horowitz and Myers go on to propose two conjectures, which are as follows:

1. SUGRA solutions: $g_{\mu \nu }=\tilde{g}_{\mu \nu }$ is the only solution to $D=10$ type IIB SUGRA, taking $\tilde{g}_{\mu \nu }\times S^{5}_{l}$ for which $\mathcal{E}= 0$, and $\mathcal{E}\geq 0$ in general.
2. Geometric solutions: $g_{\mu \nu }=\tilde{g}_{\mu \nu }$ is the only solution to $D=5$ field equations with $\Lambda = -6/l^{2}$ so that $\mathcal{E}=0$, and $\mathcal{E}\geq 0$ in general.

Seemingly, these are still open problems. However, I believe that resolving at least some aspects of these conjectures will go a long way in making sense of the geometric description of AdS/CFT.