Existence of plethystic $\mathfrak{sl}_2$ crystal seeds

Let $n\ge 0$ and let $S(n)$ be a seed. A seed is a solution to Problems 1, 2, and 3 when it satisfies the three conditions specified there; for each $r\ge n$, Definition of the crystal associated with $S(n)$ gives an operator on the set $\operatorname{SSYT}_2(1^n[r])$ of semistandard Young tableaux involved in the construction.

Seed-existence conjecture. For all $n\ge 0$, there exists a seed $S(n)$ which is a solution to Problems 1, 2, and 3, and therefore the crystal definition depending on $S(n)$ defines a crystal operator on $\operatorname{SSYT}_2(1^n[r])$ for all $r\ge n$.

This conjecture would establish the existence of the proposed plethystic $\mathfrak{sl}_2$ crystal operators in every rank parameter $n$. The paper shows the conjecture for $n\le 4$; existence for all $n$ remains open.