Rank-one derivation conjecture for non-exponentially-wild coordinates

Let $k$ be a field of characteristic zero and let $f$ be a wild coordinate of $k[\bf x]$ over $k$ that is not exponentially wild. Here $\operatorname{rank}D$ denotes the rank of a locally nilpotent derivation, and $D$ is tamely triangularizable if it becomes triangular after a tame coordinate change.

Rank-one derivation conjecture. There always exists $D\in\operatorname{LND}_k k[\bf x]$ with $\operatorname{rank}D=1$ such that $D(f)=0$ and $D$ is tamely triangularizable.

This asks whether every wild coordinate that is not exponentially wild is nevertheless killed by a particularly simple, tamely triangularizable derivation.