Equivalence of ribbon twist-braided module categories for inner-equivalent involutions

Let $\mathfrak{u}$ be a compact semisimple Lie algebra with $*$-compatible Chevalley generators $\{e_r,f_r,h_r\mid r\in I\}$ of $\mathfrak{g}=\mathfrak{u}^{\mathbb{C}}$. Let $\sigma$ be an involution of $\mathfrak{u}$, with fixed-point Lie algebra $\mathfrak{k}_{\sigma}=\mathfrak{u}^{\sigma}$; let $(X,\tau)$ be a Satake diagram with associated involution $\theta=\theta(X,\tau)$ and $\mathfrak{k}_{\theta}=\mathfrak{u}^{\theta}$; and let $(Y,\mu)$ be a Vogan diagram with associated involution $\nu=\nu(Y,\mu)$ and $\mathfrak{k}_{\nu}=\mathfrak{u}^{\nu}$. Let $0<q<1$ and $\hbar\in i\mathbb{R}$ satisfy $e^{\pi i\hbar}=q$. Main conjecture. If $\sigma$, $\theta$, and $\nu$ are inner equivalent, then

$$\operatorname{Rep}_{\hbar}(\mathfrak{k}_{\sigma})\cong \operatorname{Rep}_q^{\theta}(\mathfrak{k}_{\theta})\cong \operatorname{Rep}_q^{\nu}(\mathfrak{k}_{\nu})$$

as twist-braided $\operatorname{Rep}_q(\mathfrak{u})$-module $C^*$-categories. The conjecture asserts that the three ribbon twist-braided $\operatorname{Rep}_q(\mathfrak{u})$-module $C^*$-categories constructed in the paper depend, up to equivalence, only on the inner-equivalence class of the associated involution; the asserted equivalence remains to be established in general.