Random-matrix beyond square-root cancellation conjecture for symmetric-power characters

Let $U(N)$ be the unitary group, let $\nu_N$ be its Haar probability measure, and let $\operatorname{Sym}^k\mathbb C^N$ be the $k$th symmetric power of the standard representation. Write $\operatorname{tr}(A,\operatorname{Sym}^k\mathbb C^N)$ for its character at $A\in U(N)$. Random-matrix conjecture. Fix $B>0$. If $0\leq k\leq BN$ and $k\to+\infty$, then

$$\int_{U(N)}\left|\operatorname{tr}(A,\operatorname{Sym}^k\mathbb C^N)\right|\,d\nu_N(A)=o(1).$$

This is proposed as the random-matrix analogue of Harper's conjecture, with character twists of partial sums corresponding roughly to symmetric-power characters. Its resolution is left open in the source.