The coproduct conjecture for the algebras $\mathcal{A}_{M,N}$

Let $M,N_1,N_2\in\mathbb{Z}$. The algebras $\mathcal{A}_{M,N}$ contain a subalgebra $\mathcal{K}_{M,N}$ extending the commuting quantum toroidal $\mathfrak{gl}_1$ algebras, and $\hat\otimes$ denotes completion of the tensor product with respect to the homogeneous grading. Let $\mathcal{R}=R\check R$ be the product of the universal $R$ matrices of the two quantum toroidal subalgebras. The coproduct conjecture. There exists a homomorphism

$$\Delta_{N_1,N_2}:\mathcal{A}_{M,N_1+N_2}\longrightarrow\mathcal{A}_{M,N_1}\hat\otimes\mathcal{A}_{M,N_2}$$

which is coassociative, restricts to the standard coproduct on $\mathcal{K}_{M,N_1+N_2}$, and satisfies

$$\Delta_{N_1,N_2}X_i^+(z)=X_i^+(z)\otimes1+\mathcal{R}^{-1}(1\otimes X_i^+(z))\mathcal{R},$$ $$\Delta_{N_1,N_2}X_i^-(z)=\mathcal{R}^{-1}(X_i^-(z)\otimes1)\mathcal{R}+1\otimes X_i^-(z).$$

This would in particular make $\mathcal{A}_{M,0}$ a Hopf algebra and provide a coproduct-compatible structure for the family of extensions of commuting quantum toroidal algebras. The conjecture is proposed in the paper and no resolution is given.