Quantized universal enveloping algebra conjecture for the Neumann boundary condition

Let $\mathfrak{g}$ be a Lie bialgebra with cobracket $\delta$, let $U(\mathfrak{g})$ denote its universal enveloping algebra, and let $A^{!}_{\mathcal{N}}$ be the $E_1$ Koszul dual of the boundary algebra under the $\mathcal{N}$ boundary condition. Quantized universal enveloping algebra conjecture. The $E_1$ Koszul dual of the boundary algebra, under the $\mathcal{N}$ boundary condition, is a quantized universal enveloping algebra $U_{{\mathchar'26\mkern-9muh}}(\mathfrak{g},\delta)$ of the Lie bialgebra $(\mathfrak{g},\delta)$. The conjecture identifies the all-orders quantum deformation with the QUE quantization predicted by the leading co-Poisson structure; existence of such quantizations is attributed in the supplied context to Etingof and Kazhdan, but this identification remains unresolved here.