Strong Schur log-concavity conjecture for arithmetic progressions of Schur functions

Let $\lambda$ be a partition and let $\alpha$ be an integral vector. Arithmetic-progression Schur log-concavity conjecture. The sequence

$$s_\lambda,\quad s_{\lambda+\alpha},\quad s_{\lambda+2\alpha},\quad s_{\lambda+3\alpha},\quad\ldots$$

is strongly Schur log-concave for every such $\lambda$ and $\alpha$. The source presents this as the symmetric-function consequence of the preceding conjecture of Lam, Postnikov, and Pylyavskyy; no proof or resolution is supplied here.