Higher additive energies on discrete cubes

Let $m,n\ge 2$ be integers. We study the least exponent $t_{m,n}$ such that the $m$-fold additive energy of any subset $A$ of the discrete cube $\{0,1,\cdots ,n-1{\}}^d$ in any dimension $d$ satisfies $E_m(A)\le \mathrm{\mid }A{\mathrm{\mid }}^{t_{m,n}}$. For every fixed $m$, we obtain the asymptotic formula $$t_{m,n}=2m-1-(1+o(1)){\log }_n\frac{(2m-1{)}^{m-1\mathrm{/}2}}{(2m{)}^{m-1}}(n\to \mathrm{\infty })\mathrm{.}$$ For $m=2$, this gives $$t_{2,n}=3-(1+o(1)){\log }_n\frac{3\sqrt{3}}{4},$$ proving a conjecture of the first author.