The principal-plinth-ideal conjecture for exponential automorphisms

Let $k$ be a field of characteristic $p>0$, let $k[\boldsymbol{x}]=k[x_1,x_2,x_3]$, and let $\mathcal{E}_3(k)$ be the set of exponential automorphisms. For $\phi\in\mathcal{E}_3(k)$, write $k[\boldsymbol{x}]^\phi$ for its invariant ring and let $\operatorname{pl}(\phi)$ be the plinth ideal in $k[\boldsymbol{x}]^\phi$.

Principal-plinth-ideal conjecture.

$$k[\boldsymbol{x}]^\phi\simeq k[\boldsymbol{x}]$$

if and only if $\operatorname{pl}(\phi)$ is a principal ideal of $k[\boldsymbol{x}]^\phi$. The conjecture is supported by the theorems cited immediately before it in the source, but remains open as stated.