Sundaram's conjecture on tableaux with major index divisible by n

Let $\lambda\vdash n$ with $n\geq 1$, and let $a_{\lambda,0}$ denote the number of standard Young tableaux of shape $\lambda$ whose major index is congruent to $0$ modulo $n$. Sundaram's conjecture. For $n>1$, $a_{\lambda,0}$ is positive except in the following cases, when it is zero:

$$\lambda=(n-1,1),\qquad \lambda=(2,1^{n-2})\text{ when }n\text{ is odd},\qquad \lambda=(1^n)\text{ when }n\text{ is even}.$$

This conjecture concerns the occurrence of the trivial representation in the restriction of the Specht module $S^\lambda$ to the cyclic group generated by an $n$-cycle, equivalently the existence of tableaux with the specified modular major index. Its resolution status is not established by the supplied context.