Bijection conjecture for pseudo-involutions fixing stable root spaces

Let $A$ be the generalized Cartan matrix with Kac–Moody algebra $\mathfrak{g}$, and let $(X,\tau,\chi)$ be an enriched compatible decoration, where $Q_X$ is the root lattice associated with $X$. Consider the assignment $(X,\tau,\chi) \mapsto \theta(X,\tau,\chi)$ and the actions of $\operatorname{Aut}(A)$ on decorations and $\operatorname{Aut}(\mathfrak{g})$ on pseudo-involutions.

Bijection conjecture. The assignment $(X,\tau,\chi) \mapsto \theta(X,\tau,\chi)$ induces a bijection

$$\overline{\theta}_{\mathsf{fix}}:\left\{\text{enriched compatible decorations }(X,\tau,\chi)\text{ such that }\chi|_{Q_X}=1\right\}/\operatorname{Aut}(A)\longrightarrow\left\{\text{pseudo-involutions of the second kind fixing pointwise all stable root spaces}\right\}/\operatorname{Aut}(\mathfrak{g}).$$

This conjecture proposes that restricting to pseudo-involutions fixing pointwise all stable root spaces removes the failure of the corresponding unrestricted assignment to be injective, while the preceding corollary establishes the relevant conjugacy classification. It remains open in the source.