High-dimensional envy-free convex partition conjecture

Let $n,d$ be positive integers, and let $(\mu^1_1,\dots,\mu^1_n),\dots,(\mu^d_1,\dots,\mu^d_n)$ be $d$ tuples of $n$ absolutely continuous probability measures on $\mathbb{R}^d$. An envy-free partition for $(\mu^r_1,\dots,\mu^r_n)$ is a partition $(C_1,\dots,C_n)$ of $\mathbb{R}^d$ into convex regions such that, after assigning one region to each of the $n$ agents, every agent weakly prefers their assigned region to every other region according to their measure. High-dimensional envy-free convex partition conjecture. There exists a convex partition $(C_1,\dots,C_n)$ of $\mathbb{R}^d$ that is an envy-free partition for $(\mu^r_1,\dots,\mu^r_n)$ for each $r=1,\dots,d$. The preceding theorem establishes the corresponding result for prime-power $n$; the conjecture asks for the full general-$n$ statement, and coordination across all subdivisions is the remaining difficulty.