Upper bound on the trace of k-th Frechet derivative of the matrix function $|x^k|$

Question

For any Hermite function $A$, we define $|A|=\sqrt{A^2}$. Let $B$ be a Hermite matrix. Define $g(t)=Tr |(A+tB)^k|$.

Is it true that $|g^{(k)}(0)|$ is upper bounded by $O(\|B\|_k^k)$, where the hidden constant is independent of $A$ and the dimension when $A$ is invertible? Here $\|B\|_k$ is the Schatten k-norm of $B$.

It is easy to verify that $g^{(k)}(0)$ exists when $A$ is invertible. It is trivial when $k$ is even. It is also not hard to prove the case $k=1$.