The restricted density polynomial Hales–Jewett conjecture

For $d\in\mathbb N$ and $\delta\in(0,1)$, let $\mathcal P(\{1,\ldots,N\}^d)$ denote the power set of $\{1,\ldots,N\}^d$.

Restricted density polynomial Hales–Jewett conjecture. There exists $N=N(d,\delta)\in\mathbb N$ such that, for every $S\subseteq\mathcal P(\{1,\ldots,N\}^d)$ satisfying

$$|S|>\delta 2^{N^d},$$

there are an $A\in S$ and a non-empty $\gamma\subseteq\{1,\ldots,N\}$ such that

$$A\cap\gamma^d=\emptyset\qquad\text{and}\qquad A\cup\gamma^d\in S.$$

The paper calls this a restricted form of the density polynomial Hales–Jewett conjecture and discusses it as a possible source of positive recurrence results; no resolution is given in the supplied text.