Converse CDE conjecture for ordinary Young diagrams

Let $\lambda$ be a Young diagram. The interval $[\emptyset,\lambda]$ is the lower interval in Young's lattice, and the CDE property is the coincidence of the expected down-degree and up-degree statistics with the corresponding rank-uniform averages. A Young diagram is balanced when the left cell of every inner corner lies on the line joining the top right corner of the last cell in its first row to the bottom left corner of the last cell in its first column; the slope may be any value.

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

The forward implication for balanced diagrams is known, while the converse was checked for all partitions of size at most $30$ and remains open in general.