The absolutely continuous coupling conjecture for the leftover regions

For each $d\in\mathbb{N}$, let $\mathcal{A}$, $\mathcal{C}$, and $\mathcal{E}$ be the regions defined in the paper, and let $\operatorname{vol}_{d-1}$ denote $(d-1)$-dimensional volume. A distribution on $\mathcal{C}$ is absolutely continuous if it has a density with respect to Lebesgue measure.

Absolutely continuous coupling conjecture. For every $d\in\mathbb{N}$, there exist distributions $f_1,f_2,f_3,f_4$ on $\mathcal{C}$, absolutely continuous with respect to Lebesgue measure, such that the triples

$$\mathopen{}\mathclose\bgroup\originalleft(\operatorname{Unif}(\mathcal{A}),f_1,f_2\aftergroup\egroup\originalright)$$

and

$$\mathopen{}\mathclose\bgroup\originalleft(f_3,f_4,\operatorname{Unif}(\mathcal{E})\aftergroup\egroup\originalright)$$

are compatible, and

$$\operatorname{vol}_{d-1}(\mathcal{A})[f_1(v)+f_2(v)]+\operatorname{vol}_{d-1}(\mathcal{E})[f_3(v)+f_4(v)]<1$$

for every $v\in\mathcal{C}$.

This is the second continuous-coupling ingredient identified as sufficient for the paper's general-dimensional reduction. The supplied text gives no resolution, so the conjecture remains open.