Connectivity conjecture for the square-root queer crystal on set-valued decomposition tableaux

Let $n$ be a positive integer, and let $\mathsf{SetDecTab}_n(\lambda)$ be the set-valued decomposition tableaux of strict partition shape $\lambda$ with all entries at most $n$, equipped with the square-root queer crystal operators. A tableau is square-root queer highest weight when it is annihilated by all the corresponding raising operators.

Connectivity conjecture. For each strict partition $\lambda$ with $\ell(\lambda)\leq n$, the square-root queer crystal $\mathsf{SetDecTab}_n(\lambda)$ is connected and the tableau $T^{\mathsf{highest}}_\lambda$ is its unique square-root queer highest weight element.

This conjecture is supported by all computations reported in the paper. The claimed connectivity and uniqueness remain open in the source.