Simple-spectrum conjecture for the quantised branching algebra

Let $\sigma$ be an involution of ${\mathfrak g}$, let $H_1,\ldots,H_l$ be the chosen basic invariants with degrees $d_1,\ldots,d_l$, and let $\varpi$ be the symmetrisation map. Let $V$ be a finite-dimensional simple ${\mathfrak g}$-module, and let $V^{{\mathfrak n}_0}$ be the subspace of highest-weight vectors for ${\mathfrak g}_0$. Define $\widehat{\tilde{\mathcal Z}}\subset{\EuScript U}({\mathfrak g})$ to be generated by $\varpi((H_j)_{(i,d_j-i)})$ for $1\leq i\leq l$, $0\leq i\leq d_i$, together with ${\EuScript U}({\mathfrak g}_0)^{{\mathfrak g}_0}$. Simple-spectrum conjecture. The algebra $\widehat{\tilde{\mathcal Z}}$ acts on $V^{{\mathfrak n}_0}$ diagonalisably and with a simple spectrum. This would give a multiplicity-free spectral description of the quantised branching algebra on the highest-weight-vector space; the source states it as a conjectural goal and gives no proof or resolution.