Isolation or a positive-dimensional fixed subvariety in characteristic p

Let $F$ be a field of characteristic $p$, let $\varphi$ be an analytic map from the open unit polydisk in $n$ dimensions to itself, and let $x=(0,\ldots,0)$ be a fixed point whose multipliers are all indifferent. Isolation-or-fixed-subvariety conjecture. Either $x$ is isolated as a periodic point of $\varphi$, or, for some iterate $\varphi^{k}$, there exists a pointwise fixed subvariety of positive dimension passing through $x$. The conjecture is motivated by the characteristic-$p$ behavior that nonlinear low-degree terms of iterates can vanish after taking $p$th iterates. If true, the source notes that it would imply isolation for every fixed point with nonrepelling multipliers.