Conjecture on bracket width of vector fields on non-rational smooth plane affine curves

Let $C$ be a non-rational smooth affine plane curve, and let $\operatorname{Vec}(C)$ denote its Lie algebra of algebraic vector fields. The bracket width of a Lie algebra is the least integer $n$ such that every element is a sum of at most $n$ Lie brackets.

Bracket-width conjecture. The bracket width of $\operatorname{Vec}(C)$ is exactly two.

This conjecture would remove the unique-place-at-infinity hypothesis from the preceding result, which proves the same conclusion for non-rational smooth plane affine curves with a unique place at infinity.