Collapse conjecture for bounded-active-coordinate Hales–Jewett numbers

Let $\mathrm{HJ}^{[K]}(t,r)$ be the least dimension forcing a monochromatic combinatorial line with at most $K$ active coordinates, and let $\mathrm{HJ}(t,r)$ be the ordinary Hales–Jewett number.

Collapse conjecture.

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

for all $t\geq 2$ and $r\geq 2$.

Thus, even though dimension $\mathrm{HJ}(t,r)$ forces a monochromatic line without an active-coordinate restriction, the conjecture predicts that no uniformly bounded threshold exists when the allowed number of active coordinates is fixed at $\mathrm{HJ}(t,r)-1$. The supplied source context gives no resolution of this claim.