Polynomial Density Hales–Jewett conjecture

For positive integers $k$ and a positive real number $0<\delta<1$, define

$$K[n]=[k]^n\times[k]^{n^2}\times\dots\times[k]^{n^d}.$$

For $a\in K[n]$, let $\gamma\subset[N]$ and interpret $\gamma^r$ as the corresponding $r$-fold Cartesian product, while $\oplus$ denotes the coordinatewise insertion operation used in the source. Polynomial Density Hales–Jewett conjecture. There exists a positive integer $N(k,\delta)$ such that, for every $n\geq N(k,\delta)$, whenever $A\subset K[n]$ satisfies

$$|A|\geq\delta\cdot k^{n+n^2+\dots+n^d},$$

there exist $a\in K[n]$ and $\gamma\subset[N]$ such that

$$\{a\oplus x_1\gamma\oplus x_2(\gamma\times\gamma)\oplus\dots\oplus x_d\gamma^d:1\leq x_i\leq k,\ 1\leq i\leq d\}\subset A.$$

This is stated as a natural polynomial generalization of the density Hales–Jewett theorem and the polynomial density Hales–Jewett theorem; the source does not report a resolution.