The generalized Onsager presentation conjecture

Let $(X,\tau)\in\operatorname{GSat}(A)$ and let $\boldsymbol\gamma\in\Gamma$. Let $\mathfrak k$ be the Lie subalgebra defined by the generators and relations in the paper, including the elements corresponding to $\mathfrak h$, $\mathfrak h^\theta$, and the elements $b_{\boldsymbol i}$. Let $\widetilde{\mathfrak k}$ be generated by symbols $\widetilde h_i$, $\widetilde e_i$ for $i\in X$, $\widetilde{h_i-h_{\tau(i)}}$ for $i\in I^*$ with $i\ne\tau(i)$, and $\widetilde b_i$ for $i\in I$, subject to the relations obtained from the defining relations of $\mathfrak k$ by adding tildes.

Generalized Onsager presentation conjecture. The Lie algebra $\widetilde{\mathfrak k}$ is isomorphic to $\mathfrak k$.

This would provide a presentation of the generalized coideal Lie algebras directly from the displayed generators and relations. The source presents the assertion as a conjecture; no general proof is supplied.