The coproduct estimate conjecture for twisted Yangian generators

Let ${^Cimath\mathscr{Y}}$ be regarded as a right coideal subalgebra of the Yangian ${\mathscr{Y}}$. For generating series

$$h_i(u)=1+\sum_{r\geqslant 0}h_{i,2r+1}u^{-2r-2},$$

let $\xi_i(u)$ be the corresponding Yangian Cartan generating series, let $Q_+$ denote the positive root lattice, and let $\Delta$ be the coproduct. Twisted Yangian coproduct estimate conjecture. For a simple Lie algebra $\mathfrak{g}$, the estimates

$$h_i(u)\equiv \xi_i(u)\xi_i(-u) \pmod{{\mathscr{Y}}_{Q_+}^{\geqslant 0}[\![u^{-1}]\!]},$$ $$\Delta(h_i(u))\equiv h_i(u)\otimes\xi_i(u)\xi_i(-u) \pmod{{^\imath\mathscr{Y}}\otimes {\mathscr{Y}}^{\geqslant 0}_{Q_+}[\![u^{-1}]\!]}$$

hold. The statement is a stronger form of the preceding theorem in the case $\mathfrak{g}=\mathfrak{sl}_2$; the source indicates that this case is proved in an appendix, while the corresponding stronger estimate was previously conjectured for affine $\imath$-quantum groups of split type $\mathsf A$.