Strong Schur log-concavity conjecture for growing partition shapes

Let $\lambda, \beta$ be partitions and let $\alpha$ be an integral vector. Suppose $ell(\lambda) gg ell(\alpha)$ and $\lambda_{ell(\lambda)} gg \beta_1$. Growing-shape strong Schur log-concavity conjecture. The sequence

$$s_{\lambda},\quad s_{lambdacupbeta+\alpha},\quad s_{lambdacup^2\beta+2\alpha},\quad s_{lambdacup^3\beta+3\alpha},\quad\ldots$$

is strongly Schur log-concave. This is a precise instance of the paper's proposed general Schur-log-concavity phenomenon; the stated claim gives no resolution beyond the results proved for related special cases.