Falling-by-twos shifted Young-lattice CDE conjecture

Let $bb=(\ell,\ell-2,\ell-4,\ldots,\ell-2k)$ be a strict partition, with $ell greater than or equal to 1$ and $0\leq k<\ell/2$. Let $[\varnothing,\lambda]_{\operatorname{shifted}}$ be the interval below $bb$ in the shifted Young's lattice.

Falling-by-twos conjecture. The interval $[\varnothing,\lambda]_{\operatorname{shifted}}$ is CDE, and

$$\mathbb{E}(X)=\mathbb{E}(Y)=\frac{|\lambda|}{\ell+1}.$$

This is one of two proposed families of CDE intervals in the shifted Young's lattice; the claim is stated independently of the parity of $ell$.