Folklore extremal conjecture for sum-free subsets of higher-dimensional grids

Let $[n]=\{1,\dots,n\}$, let $[n]^d$ carry coordinatewise addition, and let $M([n]^d)$ be the maximum size of a sum-free subset. Let $k_d$ be the maximal volume of a region in the unit cube $[0,1]^d$ bounded between a hyperplane $H$ and its dilate $2\cdot H$.

Higher-dimensional extremal conjecture.

$$M([n]^d)=(k_d+o(1))n^d.$$

The conjecture is known for $d\in\{3,4,5\}$, while it remains open for larger dimensions; a stronger proposed error term is not part of the conjecture recorded here.