The lattice W-algebra centralizer conjecture

Let $\mathcal{W}'_{\mathfrak{q},N}$ be the associative algebra in the XXZ representation generated by $W\!B^{\alpha}_j$, with $1\leq j\leq N-2p+2$ and $\alpha\in\{0,\pm\}$, together with $e_k$, with $1\leq k\leq N-1$. Let $\mathcal{W}_{\mathfrak{q},N}$ denote the centralizer of $\overline{U}_{\mathfrak{q}}s\ell(2)$. Lattice W-algebra centralizer conjecture. The algebra $\mathcal{W}'_{\mathfrak{q},N}$ is isomorphic to $\mathcal{W}_{\mathfrak{q},N}$. The claim is known for $N\leq 2p-1$ and for all $N$ when $p=2$; the general case is proposed to be proved elsewhere.