Tame-coordinate conjecture for exponentials of locally nilpotent derivations

Let $k$ be a field of characteristic zero, let ${\bf x}=(x_1,x_2,x_3)$, and let $D\in\operatorname{LND}_k k[\bf x]$ be nonzero. A coordinate is tame if it is the image of a variable under a tame automorphism.

Tame-coordinate conjecture. If $\exp D$ is tame, then $D$ kills a tame coordinate of $k[\bf x]$ over $k$.

The source states that this is equivalent to the tame triangularization conjecture, so it is another formulation of the same open problem.