Preperiodic points and zero sets of commuting formal power series

Let $p$ be a prime, let $d\geq 1$, and write $X=(x_1,\ldots,x_d)$. Let $\operatorname{Preper}(u)$ denote the preperiodic points of $u$, and let $\Lambda(f)$ denote the zero set of $f$. Preperiodic-zero-set conjecture. If $u,f\in\mathbb{Z}_p[[X]]^d$ are respectively invertible and noninvertible power series without constant term, satisfy

$$u\circ f=f\circ u,$$

and $u$ and all its iterates have finitely many fixed points, then

$$\operatorname{Preper}(u)=\Lambda(f).$$

This generalizes the preceding two-dimensional result and the analogous one-variable results. The paper identifies the difficulty in proving the higher-dimensional assertion as the lack of sufficiently effective control of Newton copolygons, so the conjecture remains open.