Slope subalgebra conjecture for finite-type quantum loop groups

Fix a quantum loop group $U_q(L\mathfrak{g})$ associated with a finite-type symmetrisable Cartan matrix, where $\mathfrak{g}$ is the corresponding semisimple Lie algebra. For $\mathbf{m}\in\mathbb{R}^n$, let $\mathfrak{g}_{\mathbf{m}}$ be the semisimple Lie subalgebra of $\mathfrak{g}$ generated by the $\mathfrak{sl}_2$-triples corresponding to lattice points $\bm{\alpha}$ satisfying $\mathbf{m}\cdot\bm{\alpha}\in\mathbb{Z}$. Let $\mathcal{B}_{\mathbf{m}}$ be the slope subalgebra of $U_q(L\mathfrak{g})$.

Slope subalgebra conjecture. The slope subalgebra $\mathcal{B}_{\mathbf{m}}$ is isomorphic to the quantum group

$$\mathcal{B}_{\mathbf{m}}\cong U_q(\mathfrak{g}_{\mathbf{m}}).$$

The conjecture identifies slope subalgebras with quantum groups of semisimple Lie subalgebras determined by $\mathbf{m}$. The source states that this description has been proved using the argument from the quantum toroidal case.