Asymptotic normality conjecture for major indices of general skew tableaux

Let $\lambda/\nu^{(N)}$ be a sequence of skew shapes, let $\aft(\lambda/\nu^{(N)})$ denote the paper's auxiliary parameter, and consider the major-index distributions of standard tableaux of these skew shapes.

General skew-shape asymptotic normality conjecture. Asymptotic normality for general skew shapes, rather than only block diagonal skew shapes, holds if and only if

$$\aft(\lambda/\nu^{(N)})\to\infty\quad\text{as }N\to\infty.$$

This conjecturally characterizes exactly when the asymptotic normality result extends from block diagonal skew shapes to general skew shapes. The statement remains open.