Unfairness-witness conjecture for symmetric two-mode mixtures

Let $N_-$ be the number operator of the relative mode of a two-mode system. Consider a state

$$\Phi=\sum_i P_i\rho_i\otimes\rho_i,$$

where each $\rho_i\succeq0$ is positive semidefinite, $P_i\geq0$, and $\sum_iP_i=1$. Let $\lambda\in[-1,1]$.

Unfairness-witness conjecture. The following inequality holds:

$$\operatorname{Tr}[\Phi\lambda^{N_-}]\operatorname{Tr}[\Phi\lambda^{N_--2}N_-(N_--1)]\geq\operatorname{Tr}[\Phi\lambda^{N_--1}N_-]^2.$$

If true, the inequality gives a sufficient test for whether a two-mode state can be a convex combination of identical states in the two modes: violating it certifies that the distribution is “unfair,” although passing the test does not guarantee fairness. The conjecture is presented as preliminary numerical evidence and is open in the source.