The odd-shift inclusion conjecture for symbol components

Let $\mathcal{P}_m$ and $\mathcal{P}^{m+ke}$ be the sets appearing in the crystal-isomorphism construction, and let $X\in\mathcal{P}_m$ and $k\in\mathbb{N}$. Define

$$(X_1,X_2)=\Psi_{(e,(0,ke))}\circ\ldots\circ\Psi_{(e,(0,e))}\circ\Psi_{(e,(0,0))}(X,X)\in\mathcal{P}^{m}\times\mathcal{P}^{m+ke}.$$

Odd-shift inclusion conjecture. If $k$ is odd, then $X_1\subset X_2$.

This is the paper's main combinatorial conjecture concerning the iterated crystal isomorphisms $\Psi_{(e,(0,je))}$. The supplied text gives no evidence that it has been proved or disproved.