The conjugacy-to-triangularity conjecture for tame 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 and $\operatorname{T}_3(k)$ the tame subgroup. An automorphism $\tau$ is triangular if $\tau(x_i)\in k[x_1,\ldots,x_i]$ for each $i$.

Conjugacy-to-triangularity conjecture. For every $\phi\in\mathcal{E}_3(k)\cap\operatorname{T}_3(k)$, there exists $\sigma\in\operatorname{T}_3(k)$ such that

$$\sigma\circ\phi\circ\sigma^{-1}$$

is triangular. The conjecture is stated for characteristic zero and is presented as an open conjecture; the source suggests an analogous positive-characteristic statement.