Refined interval separation conjecture for even partitions

Let $A\in\mathcal{SP}_N$ with associated partition $\phi(A)=\lambda$. For a partition $\mu=(\mu_1\geq\cdots\geq\mu_m>0)$, define

$$\mu\!\!\downarrow=(\mu_1-1,\mu_2-1,\dots,\mu_m-1).$$

Say that $\mu$ is $\downarrow$-invertible when $\mu_{m-1}>\mu_m=1$; define $A\!\!\downarrow$ by subtracting $1$ from every part of every partition in $A$. Let $\lambda^{\dagger}$ and $\lambda^{++}$ be the partitions defined in the surrounding construction, and let separation on an interval have the meaning that the symmetric function has positive pairing with $s_A$, nonpositive pairing with the other relevant products, and support in that interval. Refined interval separation conjecture. Suppose $A\in\mathcal{SP}_N$ with $\phi(A)=\lambda$ and $\lambda$ even. If $\lambda$ is $\downarrow$-invertible and $A\!\!\downarrow\in\mathcal{SP}_{\lambda\downarrow}$, then there is a symmetric function $f$ which separates $A$ on $[\lambda,\lambda^{\dagger}]$. Otherwise, there is a symmetric function $f$ which separates $A$ on $[\lambda,\lambda^{++}]$. This is presented as a strengthening of the preceding interval-separation conjecture, potentially giving a smaller separating interval in the invertible case; no proof or resolution is supplied in the paper.