Bialgebra categorification conjecture for the multiplicity-$\alpha$ preprojective stack

Fix $\alpha\geq1$, and let $\mathcal{A}_{\Pi_{(A_{2},\alpha)}}$ be the graded algebra associated with the preprojective construction for the quiver of type $A_{2}$ with multiplicity $\alpha$. Let $\mathcal{A}_{\Pi_{(A_{2},\alpha)}}^{0}$ denote its degree-zero part, and let $\tilde{\mathfrak{n}}[u]$ be the polynomial-current Lie algebra appearing in the construction. Bialgebra categorification conjecture. There is a geometrically meaningful (co)multiplication on $\mathcal{A}_{\Pi_{(A_{2},\alpha)}}$ restricting to the existing (co)multiplication on $\mathcal{A}_{\Pi_{(A_{2},\alpha)}}^{0}$, and there is an isomorphism of bialgebras

$$\mathcal{A}_{\Pi_{(A_{2},\alpha)}}\simeq U(\tilde{\mathfrak{n}}[u]).$$

The source explicitly describes this as a heuristic conjecture for future work; no proof or disproof is given.