Take $n \in \mathbb{N}$, and consider a square matrix $A$ of size $n \times n$, with real and positive entries, and such that $\|A\|_2 \leq 1$.
I think the following statement holds from simulation, but fail to show that it is true.
$p \mapsto \lambda_i(A^p(A^T)^p)$ is non-increasing when $p$ increases, where $\lambda_i(\cdot)$ is the $i$-th largest eigenvalue (in complex norm).
For any $x\in\mathbb{R}^n$ we have $$x^TA^{p+1}(A^T)^{p+1}x = \|(A^T)^{p+1}x\|_2^2 \leq (\|A^T\|_2\|(A^T)^px\|_2)^2 \leq \|(A^T)^px\|_2^2 = x^TA^p(A^T)^px\,.$$ Using this and Courant-Fisher theorem (min-max theorem) we have \begin{align} \lambda_i(A^{p+1}(A^T)^{p+1}) &= \max_{\dim(U)=i}\;\min_{x\in U,\|x\|_2=1} x^TA^{p+1}(A^T)^{p+1}x \leq\\ &\leq \max_{\dim(U)=i}\;\min_{x\in U,\|x\|_2=1} x^TA^p(A^T)^px = \lambda_i(A^p(A^T)^p)\,, \end{align} showing your claim was correct.