Near-unimodality conjecture for major-index generating polynomials

Let $\lambda$ be a partition and let $\lambda'$ be its transpose. Let $\SYT(\lambda)^{\maj}(q)$ denote the major-index generating polynomial of standard Young tableaux of shape $\lambda$. A polynomial is nearly unimodal when its coefficients satisfy the near-unimodality condition defined in the paper.

Near-unimodality conjecture. The polynomials $\SYT(\lambda)^{\maj}(q)$ are nearly unimodal but not unimodal for partitions $\lambda$ or $\lambda'$ in the following cases: any partition of rectangle shape with more than one row and column and more than $30$ cells; any partition of the form $(k,2)$ with $k\geq4$ and $k$ even; any partition of the form $(k,4)$ with $k\geq6$ and $k$ even; or any partition of the form $(k,2,1,1)$ with $k\geq2$ and $k$ even.

This was checked for all partitions of size up to $100$, as well as 14 additional special exceptions. The conjecture concerns the precise exceptional behavior of unimodality and remains open.