The compatible-uniform-coupling conjecture for coordinate-sum slices

For $d\in\mathbb{N}$ and $t\in\mathbb{R}$, let $P_t^d$ denote the slice used in the paper and let $u_d$ maximize the volume of the coordinate-sum region defining $c_d^*$. Write $\operatorname{Unif}(P_t^d)$ for the uniform distribution on $P_t^d$. Three distributions are compatible when they can occur as the marginals of a joint distribution satisfying the required additive relation.

Compatible-uniform-coupling conjecture. For every $d\in\mathbb{N}$, the triple

$$\mathopen{}\mathclose\bgroup\originalleft(\operatorname{Unif}(P_{u_d}^d),\operatorname{Unif}(P_{u_d}^d),\operatorname{Unif}(P_{2u_d}^d)\aftergroup\egroup\originalright)$$

is compatible.

This conjecture supplies the key continuous coupling needed in the paper's reduction of the general-dimensional extremal problem. Its resolution is not given in the supplied text.