Converse CDE conjecture for shifted Young diagrams

Let $\lambda$ be a shifted Young diagram, and let $[\emptyset,\lambda]$ denote the corresponding lower interval. A shifted Young diagram is balanced according to the shifted-diagram notion used in the paper, and it is trapezoidal if it has the form $(n,n-2,\dots,n-2k+2)$. The CDE property is the coincidence of the expected down-degree and up-degree statistics with the corresponding rank-uniform averages.

Converse CDE conjecture for shifted diagrams. The interval $[\emptyset,\lambda]$ has the CDE property if and only if $\lambda$ is balanced or trapezoidal.

The balanced and trapezoidal sufficient cases are established by results cited in the source. The conjectured converse was verified for shifted Young diagrams of size at most $18$ by Hopkins and up to size $50$ in this paper, but remains open in general.