The exact-value conjecture for the two-color Hales–Jewett number

Let $HJ(k,2)$ denote the least dimension such that every coloring of $[k]^{HJ(k,2)}$ with two colors contains a monochromatic combinatorial line. Exact-value conjecture. For all $k \in \mathbb{Z}^+$, we have

$$HJ(k,2)=k.$$

The paper notes that computations for $k \in \\{2,3,4,5,6,7\\}$ support the conjecture, while the values for $k \geq 8$ were not determined there. The asymptotic result $HJ(k,2)=k(1-o(1))$ gives a close lower bound, but the exact equality remains open, particularly for composite $k$.