For any hermitian $d\times d$ matrix $A$, show that $$ \int_{\mathrm{U}(d)} UAU^\dagger d\mu(U) = \frac{A+tr(A)I}{d(d+1)}, $$ where $\mathrm{U}(d)$ is the group of unitary $d\times d$ matrices, and $d\mu$ is the Haar measure over that group.
This statement is used in equations (S34) and (S35) from Huang et al. The authors also refer to another work Gross et al (section 3.5), which apparently proves the statement using wiring diagrams. However, this work is too advanced for me. Anyway, for my purposes I only need to understand this equation for $d=2$ and $d=3$. Therefore my crude approach would be to parametrize $U, A$ and $d\mu$ in order to compute the integral explicitly. I already struggle with that and I am not sure if it is the right way to do it.
Any hint about how to show this for low dimensions would be appreciated!