Conjecture on partial symmetries in dimension three

Let ${\mathfrak D}$ denote the set of block-positive symmetries under consideration, and let $s\in{\mathfrak D}$ be a partial symmetry for $n=3$. Partial-symmetry conjecture. The rank of $s^2$ is equal to $5$, and $s$ corresponds to a decomposable positive map. The preceding examples exhibit partial symmetries of rank $5$ whose corresponding maps are decomposable, while no example with rank $7$ or with a non-decomposable corresponding map was found; the conjecture asserts that these possibilities never occur.