Let $W = aI_{n\times n} + bJ_{n\times n}$, where $I$ is an identity matix, $J$ is the matrix of all ones, $a,b\in\mathbb{R}$ and a+b>0. Also, let $A = \mathbf{P} - \mathbf{p}\mathbf{p}^{T}$, where $\mathbf{p} = (p_{1},\ldots,p_{n})^{T}$ and $\mathbf{P} = \rm{diag}(\mathbf{p})$ with $\sum_{i=1}^{n}p_i = 1$ and $0<p_i<1$. Is it possible to show that $trace((AW)^K)$ for any $K>1$ is symmteric function of $\{p_1,\ldots,p_k\}$?

I tried for $K=2,3$ and its true but in general not able to show.