The powers-of-sets containment conjecture

Let $d$ be a positive integer and let $\delta>0$. For a finite set, density means cardinality divided by the cardinality of the ambient set. The notation $[n]^d$ denotes the $d$-fold Cartesian product of $[n]$, and $\mathcal{P}([n]^d)$ denotes its power set. The powers-of-sets containment conjecture. For $n$ sufficiently large depending only on $d$ and $\delta$, every subset $\mathcal{A}\subseteq\mathcal{P}([n]^d)$ of density at least $\delta$ contains distinct subsets $A,B\subseteq[n]^d$ such that $A\subset B$ and

$$B\setminus A=S^d$$

for some subset $S\subset[n]$. This is the opposite-end specialization of the preceding conjecture when $k=2$; the source presents it as a remaining conjecture, so its general status is open.