Does there exist $p>1$ such that for all $n\geq 2$, if $(a_{ij})$ and $(b_{ij})$ are symmetric positive semidefinite $n\times n$ matrices and $a_{ij}, b_{ij}\geq 0$ then $\bigl(\|(a_{ij},b_{ij})\|_p\bigr)=\bigl((a_{ij}^p+b_{ij}^p)^{1/p}\bigr)$ is also positive semidefinite?

Maybe, a simpler question: is it true for $p=2$?

**Edited:** Original question did not have the condition $a_{ij}, b_{ij}\geq 0$. If we take $b_{ij}=0$, it is possible that $(|a_{ij}|)$ is not positive semidefinite when $(a_{ij})$ is.