Lam–Postnikov–Pylyavskyy conjecture on Schur positivity

Let $\lambda$, $\mu$, $\lambda'$, and $\mu'$ be partitions in $\mathbb{Z}^n$. For vectors $x,y\in\mathbb{Z}^n$, define

$$\Pi(x,y)=\{z\in\mathbb{Z}^n:\min(x_i-x_j,y_i-y_j)\le z_i-z_j\le\max(x_i-x_j,y_i-y_j)\text{ for all }1\le i<j\le n\}.$$

A symmetric polynomial is Schur nonnegative if it is a nonnegative linear combination of Schur polynomials. Lam–Postnikov–Pylyavskyy's conjecture. If $\lambda'+\mu'=\lambda+\mu$ and $\lambda'\in\Pi(\lambda,\mu)$, then

$$s_{\lambda'}s_{\mu'}-s_{\lambda}s_{\mu}$$

is Schur nonnegative. The paper presents a proof using skeps and $L$-log-concavity, so the conjecture is resolved.