Reiner–Tenner–Yong's trapezoidal shifted Young lattice CDE conjecture

For $n\geq1$ and $0\leq k<n/2$, let

$$\lambda:=(n,n-2,n-4,\ldots,n-2k).$$

The notation $[\varnothing,\lambda]_{\mathrm{shift}}$ denotes the corresponding initial interval in the shifted Young's lattice, $\mathrm{uni}_{[\varnothing,\lambda]_{\mathrm{shift}}}$ its uniform distribution, and $\mathrm{ddeg}$ the down-degree. Reiner–Tenner–Yong's trapezoidal shifted Young lattice conjecture. The interval $[\varnothing,\lambda]_{\mathrm{shift}}$ is CDE, with edge density

$$\mathbb{E}(\mathrm{uni}_{[\varnothing,\lambda]_{\mathrm{shift}}};\mathrm{ddeg})=\frac{|\lambda|}{n+1}.$$

This is the second conjecture of Reiner, Tenner and Yong about CDE initial intervals of the shifted Young's lattice; the supplied context does not state whether it has been resolved.