Consider Schur polynomials $s_\lambda$ with $\lambda = (2m, m, m, \ldots, m, 0)$ and $\ell(\lambda) = n$ (that is, $\lambda$ has $n$ rows). Here $m \gg n$, which, for the sake of concreteness, let's assume to be $n^2$.
Let $z_1, \ldots, z_n \in \mathbb{T}$ be $n$ complex numbers of unit length, satisfying $\sum_{j=1}^n z_j^k = O(\log n)$, for all $1 \le k \le K$ and some absolute constant $K$.
Finally let $d_\lambda = \prod_{1 \le j < k \le n} (1 + \frac{\lambda_j - \lambda_k}{k - j})$ be the dimension of the corresponding $SU(n)$ irreducible representation. How can I show that $|s_\lambda(z_1, \ldots, z_n)| \le d_\lambda^{1 - \epsilon}$, for some positive $\epsilon$, for all sufficiently large $n$, provided that $K$ is sufficiently large?
If the above problem is still too hard, I will grant full credit to solutions of the problem replacing the moment conditions on $z_1, \ldots, z_n$ with $\sum_{j=1}^n z_j^k = 0$ for all $1 \le k \le 100$ (that is, $K = 100$). In fact, even result with $\epsilon$ tending to $0$ and $K$ tending to $\infty$ will be appreciated, as long as they don't converge too fast.