Angular momentum theory problem probably wrong sign

[lgnr]

I have to prove that $\left\langle lm | \vec{\hat{r}} \times \vec{\hat{p}} | lm\right \rangle = \left\langle lm | \vec{\hat{p}} \times \vec{\hat{r}} | lm \right \rangle$, where $| lm \rangle$ are eigenkets of angular momentum operator $\hat{L}^2$

And I can't figure out a way to do this correctly. I wrote the angular momentum operator-vector in terms of its components, $\hat{L_x}$, $\hat{L_y}$ and $\hat{L_z}$, and only the $\hat{k}$ component survives the bra-ket operation, because I can write $\vec{L_x}$ y $\vec{L_y}$ in terms of ladder operators, and after lowering and rising the eigenstates, the corresponding eigenkets (except for the $\hat{k}$ component) cancel with $\langle lm |$ because of the orthogonality property of these eigenkets. Probably there is a wrong sign in the problem statement. Looks trivial in that case, but I want to know your opinion.

Thanks in advance for any advice.

 

[Bill_K]

Undoubtedly the wrong sign.