Diagonal-only coloring conjecture for Hales–Jewett lines

Let $[t]^n$ be the set of words of length $n$ over an alphabet of size $t$, let $r$ be the number of colors, and let $\mathrm{HJ}(t,r)$ be the Hales–Jewett number. The diagonal is the line whose active coordinates are all coordinates and whose constant letter is varied. Diagonal-only conjecture. For every $n\le\mathrm{HJ}(t,r)$, some $r$-coloring of $[t]^n$ has the diagonal as its only monochromatic line. The range is maximal, since the source proves that no such coloring exists for $n>\mathrm{HJ}(t,r)$.