  We prove the positive energy conjecture for a class of asymptotically
Horowitz-Myers (AHM) metrics on $\mathbb{R}^{2}\times\mathbb{T}^{n-2}$.This
generalizes the previous results of
Barzegar-Chru\'{s}ciel-H\"{o}rzinger-Maliborski-Nguyen \cite{BCHMN} as well as
the authors \cite{LZ}.
