Tame triangularization conjecture for 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 derivation is tamely triangularizable if some tame change of coordinates makes it triangular.

Tame triangularization conjecture. If $\exp D$ is tame, then $D$ is tamely triangularizable.

This conjecture seeks to characterize tame exponentials of locally nilpotent derivations in three variables; the source presents it as an open conjecture.