Conjecture on bi-shapes occurring in products of predecessor relations

Let $(\gamma|\lambda)$ be a bi-shape occurring in $(S_t)_\mu$. Suppose that there exists a $1$-predecessor $\mu'$ of $\mu$ containing a $t$-bi-predecessor $(\alpha|\beta)$ of $(\gamma|\lambda)$. Bi-shape predecessor conjecture. Then $(\gamma|\lambda)$ occurs in $(S_t)_1(S_t)_{(\alpha|\beta)}$. This conjecture formalizes the computational observation that relations associated with predecessor bi-shapes are generated by lower-degree relations. A positive answer would bring the authors closer to proving the conjecture that the specified highest weight relations generate the full ideal of relations, but the assertion itself is not proved.