The power-of-three anti-van der Waerden conjecture

Let $m$ be a nonnegative integer, and let $\operatorname{aw}([n],3)$ denote the least number of colors such that every exact coloring of $[n]=\{1,\ldots,n\}$ contains a rainbow three-term arithmetic progression.

Power-of-three anti-van der Waerden conjecture. For every nonnegative integer $m$,

$$\operatorname{aw}([3^m],3)=m+2.$$

The paper notes that this agrees with the available computed data and would make the established lower bound exact whenever $n$ is a power of three. Its general validity remains open.