Sharp sumset lower-bound conjecture for hypercubes

Let $n,m,d$ be positive integers, let $[m]:=\{0,1,\ldots,m\}$, and let $A_1,\ldots,A_n$ be finite subsets of $[m]^d$. Define

$$p_{n,m}:=\frac{n\log(m+1)}{m\log(n+1)}.$$

Sharp sumset lower-bound conjecture. The inequality

$$|A_1+\cdots+A_n|\geq (|A_1|\cdots|A_n|)^{1/p}$$

holds with $p=p_{n,m}$.

The conjecture asks for the sharp exponent in lower bounds for Minkowski sums of subsets of a hypercube. The paper states that it resolves this conjecture in full generality, so the conjecture is solved.