The compatible-triple conjecture for real forms of holomorphic inner product Lie algebras

Let $(\mathfrak{g}^{\mathbb{C}}, g^{\mathbb{C}})$ be a holomorphic inner product space, with $\mathfrak{g}^{\mathbb{C}}$ a complex Lie algebra. Let $\mathfrak{g}\subset \mathfrak{g}^{\mathbb{C}}\supset \tilde{\mathfrak{g}}$ be real forms that are also real slices, and assume that a compact real form $\mathfrak{u}\subset \mathfrak{g}^{\mathbb{C}}$ exists. A triple of Lie algebras is compatible when its members have commuting conjugation maps in the sense of the paper. Compatible-triple conjecture. There exist automorphisms $\phi_1,\phi_2,\phi_3\in \operatorname{Aut}(\mathfrak{g}^{\mathbb{C}})\cap O(n,\mathbb{C})$ such that

$$\bigl(\phi_1(\mathfrak{g}),\phi_2(\tilde{\mathfrak{g}}),\phi_3(\mathfrak{u})\bigr)$$

is a compatible triple of Lie algebras. The conjecture would extend the paper's Wick-rotation framework from Riemannian targets to non-Riemannian signatures; the source notes that it holds for abelian Lie algebras and in many examples among classical semisimple Lie algebras, but does not establish it in general.