The mixed-volume Minkowski-sum volume conjecture

Let $d$ be a positive integer, let $m$ be a real number, and let $K_1,\ldots,K_d$ be convex bodies in $\mathds{R}^d$ satisfying

$$\operatorname{Vol}(K_1) \ge 1,\ \ldots,\ \operatorname{Vol}(K_d) \ge 1,\quad\text{and }\ \operatorname{V}(K_1,\ldots,K_d)=m.$$

For an integer $\ell$ with $1\leq\ell\leq d$, the mixed-volume Minkowski-sum volume conjecture. The maximum of $\operatorname{Vol}(K_1+\cdots+K_\ell)$ equals $(m+\ell-1)^d$ and is attained when $K_1=mK_2=\cdots=mK_d$ with $\operatorname{Vol}(K_d)=1$.