The orbit-by-orbit genomic tableau inequality

Let $\mathcal{O} \subseteq \mathrm{LR}(\alpha,\scalebox{.5}{\yng(1)},\beta, \gamma)$ be an orbit of $\omega$. Let $K_1(\mathcal{O})$ and $K_2(\mathcal{O})$ denote the sets of genomic tableaux occurring in this orbit in Phases 1 and 2, respectively, via the bijections $\varphi_1$ and $\varphi_2$. Orbit-by-orbit genomic tableau conjecture. For $i=1,2$, one has

$$|K_i(\mathcal{O})| \geq |\mathcal{O}|-1.$$

Numerical evidence supports the inequality. It has been verified in Phase 1 when $\beta$ has two rows and in Phase 2 when $\beta$ has two columns; it also holds for orbits of size two, with equality for fixed points of $\omega$.