The admissible-order picture–crystal bijection conjecture

Let $\lambda,\mu,\nu$ be Young diagrams. Let $A$ be an admissible order on $\nu\setminus\lambda$ and let $A'$ be an admissible order on $\mu$. The set $\mathbf{B}(\mu)_\lambda^\nu[A']$ consists of tableaux whose admissible reading produces a sequence of Young diagrams from $\lambda$ to $\nu$, and ${\bf P}(\mu,\nu\setminus\lambda:A,A')$ is the set of bijective maps from $\mu$ to $\nu\setminus\lambda$ that are PA-standard with inverse PA$'$-standard. Admissible-order picture–crystal bijection conjecture. There exists a bijection

$$\Psi:\mathbf{B}(\mu)_\lambda^\nu[A']\longrightarrow {\bf P}(\mu,\nu\setminus\lambda:A,A'),$$

where $\Psi$ is the same as in 4.1. The assertion proposes that the correspondence between Littlewood–Richardson crystals and pictures is independent of the chosen admissible orders, extending the bijection previously defined in Section 4.1.