Cancellation equivalence for symmetric Poisson algebras and enveloping algebras

Let $\mathfrak g$ be a finite-dimensional Lie algebra. Its symmetric Poisson algebra is denoted by $PS(\mathfrak g)$, and its universal enveloping algebra by $U(\mathfrak g)$. Cancellation equivalence conjecture. $PS(\mathfrak g)$ is Poisson cancellative if and only if $U(\mathfrak g)$ is cancellative. The paper proves this for all nontrivial Lie algebras of dimension at most $3$, while the higher-dimensional case remains open.