The positivity conjecture for cyclic-intersection functions

Let $J=\{j_1,\ldots,j_k\}$ be a $k$-subset of $[n]$, and let $I_{j_\alpha}$ be the $j_\alpha$th cyclic $k$-interval. Let $\xi_J$ be the quantity defined in the source's geometric characterization theorem, and let $\bar I_{j_1}\cap\cdots\cap\bar I_{j_k}$ denote the corresponding intersection. The cyclic-intersection positivity conjecture. The function

$$\varepsilon_{J\bar J}(-1)^{\xi_J}\cdot\langle Y\,\bar I_{j_1}\cap\cdots\cap\bar I_{j_k}\rangle$$

is strictly positive, where $\varepsilon_{J\bar J}$ is the Levi-Civita symbol obtained by listing the elements of $J$ and $\bar J$ in increasing order. Such positivity would support the sign structure of the canonical-function formulas developed for amplituhedra, but the source does not provide resolution.