The invariant-coordinate conjecture for positive-characteristic tame automorphisms

Let $k$ be a field of characteristic $p>0$, let $k[\boldsymbol{x}]=k[x_1,x_2,x_3]$, let $\mathcal{C}_3(k)$ be the set of automorphisms of characteristic-order, and let $\operatorname{T}_3(k)$ be the tame subgroup. For $\phi\in\mathcal{C}_3(k)\cap\operatorname{T}_3(k)$, write $k[\boldsymbol{x}]^\phi=\{f\in k[\boldsymbol{x}]\mid\phi(f)=f\}$. Let $\gamma$ denote the invariant-ring invariant used in the source.

Invariant-coordinate conjecture. For every such $\phi$, there exists $\sigma\in\operatorname{T}_3(k)$ such that

$$\sigma(x_1)\in k[\boldsymbol{x}]^\phi.$$

Hence,

$$\gamma\bigl(k[\boldsymbol{x}]^\phi\bigr)\geq 1.$$

The source states this as a consequence of the positive-characteristic conjugacy-to-triangularity conjecture; its status is therefore open.