Gelfand–Kirillov conjecture for Lie algebras of linear algebraic groups

Let $\mathfrak{g}$ be the Lie algebra of a linear algebraic group over $\mathbb{C}$. For $r,s\in\mathbb{Z}_+$, let $\mathcal{D}_{r,s}=\mathcal{D}_r\otimes\mathbb{C}[y_1,\dots,y_s]$, and denote its skew field of fractions by $\mathbf{F}_{r,s}$. Let $\mathbf{F}(\mathfrak{g})$ denote the skew fraction field of $U(\mathfrak{g})$. Gelfand–Kirillov conjecture. There exist $r,s\in\mathbb{Z}_+$, depending on $\mathfrak{g}$, such that

$$\mathbf{F}(\mathfrak{g})\cong\mathbf{F}_{r,s}.$$

This is the formulation introduced by Gelfand and Kirillov for linear algebraic Lie algebras. The supplied status evidence says that the conjecture was settled for nilpotent Lie algebras, $\mathfrak{sl}_n$, and $\mathfrak{gl}_n$ by Gelfand and Kirillov; the general formulation is therefore recorded here as solved according to the provided evidence.