Generalized presentation conjecture for quantum symmetric pair coideals

Let $X \subseteq I$ be of finite type and let $\tau \in \operatorname{Aut}_X(A)$ satisfy $\tau^2=\operatorname{id}_I$ and $\tau|_X=\operatorname{id}_X$. For $(\boldsymbol\gamma,\boldsymbol\sigma) \in (\mathbb{F}^{\times})^I \times \mathbb{F}^I$, let $\widetilde B$ be the algebra freely generated over $U_q\mathfrak n_X^+U_q\mathfrak h^{\theta(X,\tau)}$ by elements $\widetilde B_i$, and let $\Pi:\widetilde B\to U_q\mathfrak k_{\boldsymbol\gamma,\boldsymbol\sigma}(X,\tau)$ send $\widetilde B_i$ to $B_{i;\gamma_i,\sigma_i}$ and act identically on the coefficient algebra. The elements and relations below are understood with the notation of the source, including $\lambda_{ij}$, $\operatorname{Ser}_{ij}$, and the ordered monomials $\widetilde B_{\boldsymbol i}$. Generalized presentation conjecture. There exist elements

$$\widetilde C_{ij}(\boldsymbol\gamma)\in \sum_{\alpha_{\boldsymbol i}<\lambda_{ij}}U_q\mathfrak n_X^+U_q\mathfrak h^{\theta(X,\tau)}\widetilde B_{\boldsymbol i}$$

for all $i\ne j$ such that $\ker(\Pi)$ is generated by

$$t_\lambda\widetilde B_i-q^{-(\lambda,\alpha_i)}\widetilde B_i t_\lambda,$$ $$E_i\widetilde B_j-\widetilde B_jE_i-\delta_{ij}\frac{t_i-t_i^{-1}}{q_i-q_i^{-1}},$$

and

$$\operatorname{Ser}_{ij}(\widetilde B_i,\widetilde B_j)-\widetilde C_{ij}(\boldsymbol\gamma),$$

with the quantifiers and index ranges stated in the source, if and only if $(X,\tau)\in\mathcal{G}\operatorname{Sat}(A)$ and $(\boldsymbol\gamma,\boldsymbol\sigma)\in\mathcal{G}\mathcal{S}_q$. Moreover, the $\widetilde C_{ij}(\boldsymbol\gamma)$ are independent of $\boldsymbol\sigma$. This proposes a presentation of a broader class of quantum symmetric pair coideals; the source gives no resolution of the if-and-only-if assertion.