Rank-three wildness conjecture for exponentials

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. The rank of $D$ is the least number of variables needed after a coordinate change to express $D$.

Rank-three wildness conjecture. If $\operatorname{rank}D=3$, then $\exp D$ is wild.

The source derives this conjecture from the tame-coordinate conjecture; it asserts that maximal-rank locally nilpotent derivations cannot have tame exponentials.