Let $X$ be a Poisson variety. There is a concept "cluster algebra structure compatible with Poisson structure" introduced in the paper.

Suppose that we construct a maximal independent set of functions $P_1, \ldots, P_n$ in the coordinate ring of $X$ which is log-canonical and we construct an initial quiver $Q$ such that the matrix $B$ and the matrix $\Omega$ of the Poisson brackets of $P_1, \ldots, P_n$ satisfy $B \Omega = [D \ 0]$, where $D$ is a non-degenerate diagonal matrix. Is the following true: after any sequence of mutations, the cluster variables are still log-canonical? Thank you very much.