Collapse conjecture for the Hales–Jewett bracket hierarchy

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 $\mathrm{HJ}^{[k]}(t,r)$ denote the least dimension forcing a monochromatic line in the $k$-th bracket class, with value $\infty$ if no such dimension exists. Let $\mathrm{HJ}(t,r)$ be the Hales–Jewett number, and let $\kappa_{\mathrm{all}}(t,r)$ be the largest bracket level whose line-free colorings exist in every dimension. Collapse conjecture.

$$\mathrm{HJ}^{[\mathrm{HJ}(t,r)-1]}(t,r)=\infty$$

for all $t,r\ge2$; equivalently, $\kappa_{\mathrm{all}}(t,r)=\mathrm{HJ}(t,r)-1$. This is known wherever the relevant Hales–Jewett numbers are known, but remains open in general.