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

Let $\delta_r=(r,r-1,\ldots,1)$, and for partitions let $\nu+\lambda$ denote the partition obtained by adding corresponding parts. For $a,m,n\geq 1$ with $n>am$, set

$$\lambda:=\delta_n+\delta_m\circ a^a.$$

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

$$\mathbb{E}(\mathrm{uni}_{[\varnothing,\lambda]_{\mathrm{shift}}};\mathrm{ddeg})=\frac{n+1+am}{4}.$$

This is one of two conjectures concerning CDE initial intervals of the shifted Young's lattice; the source proves this conjecture, so it is no longer open.