Commuting noninvertible formal power series conjecture

Let $p$ be a prime, let $d\geq 1$, and write $X=(x_1,\ldots,x_d)$. For a power series $h\in\mathbb{Z}_p[[X]]^d$, let $\Lambda(h)$ denote its zero set. Commuting noninvertible formal power series conjecture. If $f,g\in\mathbb{Z}_p[[X]]^d$ are noninvertible power series without constant coefficient and commute under composition, namely

$$f\circ g=g\circ f,$$

then

$$\Lambda(f)=\Lambda(g).$$

This extends the corresponding one-variable results and the proved two-variable case to arbitrary dimension. The authors state that the required control of Newton copolygons in more than two variables is currently unavailable, so the assertion remains open.