The logarithmic upper-bound conjecture for anti-van der Waerden numbers

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

Logarithmic upper-bound conjecture. There exists a constant $C$ such that

$$\operatorname{aw}([n],3)\leq\left\lceil\log_3 n\right\rceil+C$$

for all $n\geq 3$.

The paper has already established the lower bound $\operatorname{aw}([n],3)\geq\lceil\log_3 n\rceil+2$ and an upper bound of order $\log n$; this conjecture asserts that the base-three lower bound is sharp up to an additive constant.