Almost-everywhere log-concavity conjecture for major-index generating polynomials

Let $\lambda\vdash n$ be a partition, and let $\SYT(\lambda)^{\maj}(q)$ denote the major-index generating polynomial of standard Young tableaux of shape $\lambda$. A polynomial is log-concave when its coefficients $c_i$ satisfy $c_i^2\geq c_{i-1}c_{i+1}$ wherever these coefficients are defined.

Log-concavity conjecture. The polynomials $\SYT(\lambda)^{\maj}(q)$ are almost always log-concave for partitions $\lambda\vdash n$ for large $n$.

The conjecture is motivated by the log-concavity of the normal distribution and computational data showing increasing proportions of log-concave examples for $n=30,40,50$. The meaning of “almost always” is not quantified in the statement, so the precise asymptotic formulation remains open.