The shifted Yangian conjecture for cocharacter B-algebras

Let $\rho$ be a cocharacter of $SL_N$, and use the combined $\mathbb C^\times$-action on $U^{comb}_\delta(\mathscr O_\epsilon(\mathbb C^2)\otimes\mathfrak{gl}_N)$ obtained from the opposite-weight action on $\mathbb C^2$ and the adjoint action of $\rho$. Denote the resulting $B$-algebra by $\mathcal B_\rho(U^{comb}_\delta(\mathscr O_\epsilon(\mathbb C^2)\otimes\mathfrak{gl}_N))$. Shifted Yangian conjecture. There is an isomorphism

$$\mathcal B_\rho\left(U^{comb}_\delta(\mathscr O_\epsilon(\mathbb C^2)\otimes\mathfrak{gl}_N)\right)\cong Y_\rho(\mathfrak{gl}_N),$$

where $Y_\rho(\mathfrak{gl}_N)$ is the shifted Yangian associated to $\rho$. This extends the preceding Yangian conjecture to cocharacter-dependent shifts; the claimed identification relies on definitions and comparisons with shifted Yangians in the cited literature and remains unproved here.