Freeness conjecture for the symmetric algebra of a nilradical

Let $\mathfrak g$ be a simple Lie algebra, let $\mathfrak n$ be a nilradical in $\mathfrak g$, and let $U$ be the maximal unipotent subgroup associated with the chosen Borel subgroup. Write $\mathcal S(\mathfrak n)$ for the symmetric algebra of $\mathfrak n$, and let $\mathcal Z(\tilde{\mathfrak n})$ denote the Poisson centre associated with an optimal nilradical $\tilde{\mathfrak n}$. Freeness conjecture. For any nilradical $\mathfrak n$ in a simple Lie algebra $\mathfrak g$, $\mathcal S(\mathfrak n)$ is a free $\mathcal S(\mathfrak n)^U$-module. In particular, for any optimal nilradical $\tilde{\mathfrak n}$, $\mathcal S(\tilde{\mathfrak n})$ is a free $\mathcal Z(\tilde{\mathfrak n})$-module. The statement is suggested by the computations preceding it; the supplied text does not establish it in general, so its general validity remains open.