Indefinite energy minimization conjecture

Let $V$ be a vector space with a non-singular quadratic form $Q$ of signature $(a,b)$, and let

$$H^+=\{\vec{x}\in V\mid Q(\vec{x})=1\}$$

be its positive unit hyperboloid. Let $N$ be a spacelike submanifold of $H^+$ whose inclusion into $H^+$ is a homotopy equivalence. Write $\operatorname{\stackrel{\longrightarrow}{Vol}}N$ for its directed volume.

Indefinite energy conjecture. The quantity $Q(\operatorname{\stackrel{\longrightarrow}{Vol}}N)$ is uniquely minimized when $N$ is the intersection of $Q$ with an $a$-plane in $V$ containing the origin.

This is presented as a generalization of the energy conjecture to indefinite orthogonal geometry. The source proves special cases of the generalization but does not report a proof of the full statement.