Interval separation conjecture for the Schur cone

Let $\mathcal{SP}_N$ be the collection of multisets of partitions with total size $N$, let $\phi(A)=\lambda$ be the associated partition, and let $\lambda^{++}$ be defined by

$$\lambda^{++}=\begin{cases}(\lambda_1+1,\lambda_2+1,\dots,\lambda_k+1,\lambda_{k+1}-1,\dots,\lambda_m-1)&\text{if }m=2k,\\(\lambda_1+1,\lambda_2+1,\dots,\lambda_k+1,\lambda_{k+1},\lambda_{k+2}-1,\dots,\lambda_m-1)&\text{if }m=2k+1.\end{cases}$$

A symmetric function $f$ separates $A$ on $[\lambda,\rho]$ when its Schur support lies in that interval, its pairing with $s_A$ is positive, and its pairing with every other relevant $s_B$ is nonpositive. Interval separation conjecture. For every $A\in\mathcal{SP}_N$ with $\phi(A)=\lambda$, there is a symmetric function $f$ such that $f$ separates $A$ on $[\lambda,\lambda^{++}]$. This conjecture supplies separating functions on a prescribed dominance interval and is intended to establish extremality by separating each product from the others; the surrounding discussion notes that the analogous smaller interval $[\lambda,\lambda^+]$ is insufficient when parts of $\lambda$ are repeated.