Let $p_\lambda$ be power sum symmetric functions. Let $s_\lambda$ and $o_\lambda$ be irreducible characters of the unitary and orthogonal groups $U(N)$ and $O(N)$, respectively (the $s$ are the Schur functions).

Then $$p_\mu(X)=\sum_{|\lambda|=n}\chi_\lambda(\mu)s_\lambda(X)$$ and $$p_\mu(X)=\sum_{|\lambda|\leq n}b_{N,\lambda}(\mu)o_\lambda(X),$$ where $\chi_\lambda(\mu)$ are irreducible $S_n$ characters and $b_{N,\lambda}(\mu)$ are what I am calling Brauer characters (in contrast with the former, the latter depend on $N$, the dimension of the matrix $X$).

For the special case when $\mu$ is the singleton, $\mu=(n)$, it is known that $\chi_\lambda(n)$ is non-zero only if $\lambda$ is a hook, and it is $\pm 1$ depending on the size of the hook. In particular, this character is bounded, i.e. $|\chi_\lambda(n)|\leq 1$ for any $\lambda$.

My question is if the corresponding Brauer characters are also bounded, $|b_{N,\lambda}(n)|\leq 1$ for any $\lambda$ and $N$?