Braided tensor equivalence conjecture for the quantum Hamiltonian reduction functor

Let $\mathfrak{g}$ be the affine Lie algebra under consideration, let $\mathrm{KL}_k(\mathfrak{g})$ be the category generated by the modules $L_k(\lambda)$ for $\lambda\in\mathrm{Pr}_{\mathbb{Z}}^k$, and let $\mathrm{KL}_{k,f}(\mathfrak{g})$ be the category of $\mathcal{W}_k^{D^+}$-modules generated by $\mathbf{L}_k(\lambda)$. Consider the functor

$$H_f^0\colon \mathrm{KL}_k(\mathfrak{g})\longrightarrow\mathrm{KL}_{k,f}(\mathfrak{g}),\qquad L_k(\lambda)\mapsto\mathbf{L}_k(\lambda).$$

Braided tensor equivalence conjecture. The functor $H_f^0$ is an equivalence of braided tensor categories. The source already states that this functor is an equivalence of abelian categories and that both categories are braided tensor categories; compatibility with the braided tensor structures is the remaining conjectural assertion.