Symmetric-coloring extremality conjecture for Hales–Jewett numbers

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. Let $\mathrm{HJ}_{\mathrm{sym}}(t,r)$ denote the least dimension in which every symmetric $r$-coloring contains a monochromatic combinatorial line. Symmetric-coloring extremality conjecture.

$$\mathrm{HJ}_{\mathrm{sym}}(t,r)=\mathrm{HJ}(t,r)$$

for all $t,r\ge2$. The inequality $\mathrm{HJ}_{\mathrm{sym}}(t,r)\le\mathrm{HJ}(t,r)$ is immediate, and equality holds wherever the Hales–Jewett number is known; the conjecture asserts equality in general.