Fomin–Fulton–Li–Poon's Schur-positivity conjecture for sorted partitions

Let $\lambda$ and $\mu$ be partitions, and let $\lambda\cup\mu=(\nu_1,\nu_2,\nu_3,\ldots)$ be the partition obtained by rearranging all parts of $\lambda$ and $\mu$ in weakly decreasing order. Define

$$\operatorname{sort}_1(\lambda,\mu)=(\nu_1,\nu_3,\nu_5,\ldots),\qquad \operatorname{sort}_2(\lambda,\mu)=(\nu_2,\nu_4,\nu_6,\ldots).$$

For symmetric functions $f$ and $g$, $f\geq_s g$ means that $f-g$ is Schur nonnegative. Fomin–Fulton–Li–Poon's conjecture.

$$s_{\operatorname{sort}_1(\lambda,\mu)}s_{\operatorname{sort}_2(\lambda,\mu)}\geq_s s_\lambda s_\mu.$$

This conjecture concerns Schur positivity arising from eigenvalue and singular-value inequalities for sums of Hermitian and complex matrices. It was also studied by Bergeron and McNamara, but the source does not specify a resolution status.