Orbit-by-orbit inequality for genomic tableaux in the orthogonal Grassmannian

Let $\omega$ be the permutation governing the switching algorithm, and let $\mathcal{O}\in\operatorname{Orb}(\omega)$ be one of its orbits. A genomic tableau is a tableau counted by the relevant type B Littlewood–Richardson rule, and standardization and reverse standardization order refer to the two traversal procedures in the switching algorithm. Orbit-by-orbit conjecture. There are at least $|\mathcal{O}|-1$ genomic tableaux traversed in standardization order and at least $|\mathcal{O}|-1$ traversed in reverse standardization order in the computation of $\mathcal{O}$ using the switching algorithm. This refines the global inequality by requiring the contribution $|\mathcal{O}|-1$ separately for every orbit; the paper reports computer evidence but does not establish the claim.