Bijective decomposition conjecture for Schur-function Plücker relations

Let $\lambda$ be a partition with $n$ outside corners. Choose a corner $k$ and corresponding weight $\omega_\ell$, and retain the notions of Theorem~

. Let $L$ be the set of semistandard Young tableaux $\operatorname{SSYT}(\lambda)$ acting legally on $Y(\lambda)$. **Bijective decomposition conjecture.** The following hold: 1. The tableaux in $\operatorname{SSYT}(\lambda-\omega_\ell)$ which act legally on $Y(\lambda+\omega_\ell)$ can be put in bijection with a \subset $A$ of $L$. 2. There are subsets $B^i_j\subseteq L\setminus A$, for each $1\leq i\leq k\leq j\leq n$, such that $B^i_j$ is in bijection with $\operatorname{SSYT}(\mu^i_j(\lambda))$ acting legally on $Y(\pi^i_j(\lambda))$. 3. $L=A\cup\bigcup B^i_j$. 4. The intersection $B^{i_1}_{j_1}\cap\cdots\cap B^{i_r}_{j_r}$ is nonempty if and only if the terms can be reordered to satisfy

1\leq i_1<\cdots<i_r\leq k\leq j_r<\cdots<j_1\leq n,

and in that case it is in bijection with $\operatorname{SSYT}(\mu^{i_1\cdots i_r}_{j_1\cdots j_r}(\lambda))$ acting legally on $Y(\pi^{i_1\cdots i_r}_{j_1\cdots j_r}(\lambda))$. All bijections between $\operatorname{SSYT}(\lambda)$ acting on $Y(\lambda)$ and $\operatorname{SSYT}(\alpha)$ acting on $Y(\beta)$ should respect the Young diagrams produced by the two actions. The conjecture proposes a general bijective proof for the arbitrary-partition form of the Plücker relation, extending the preceding rectangular-case argument. Its precise combinatorial content depends on the notions introduced in Theorem~

; the paper presents it as a belief based on computational examples, and no resolution is supplied.