Extremality conjecture for symmetric Hales–Jewett thresholds

Let $t,r\geq 2$. Write $n_{\mathrm{sym}}(t,r)$ for the largest $n$ such that $[t]^n$ admits a line-free symmetric $r$-coloring, and let $\mathrm{HJ}(t,r)$ denote the least $n$ such that every $r$-coloring of $[t]^n$ contains a monochromatic combinatorial line. A line-free symmetric coloring is in particular line-free, so

$$n_{\mathrm{sym}}(t,r)\leq \mathrm{HJ}(t,r)-1.$$

Extremality conjecture. For all $t,r\geq 2$,

$$n_{\mathrm{sym}}(t,r)=\mathrm{HJ}(t,r)-1.$$

The conjecture asserts that the upper bound obtained by restricting to symmetric colorings is always attained. The cited constructions give $n_{\mathrm{sym}}(3,3)\geq 21$ and $n_{\mathrm{sym}}(4,2)\geq 13$, but the general equality remains open.