Reverse log-majorization for nonnegative matrix powers

Let $A,B\geq 0$, and let $p,q,r$ be real parameters satisfying $0<p\leq q$ and $r\geq 0$. Here $\succ_{\log}$ denotes log-majorization. Reverse log-majorization conjecture. One has

$$A^{r+q} B^q \succ_{\log} A^r \left( A^{p/2} B^p A^{p/2} \right)^{q/p}.$$

This is proposed as a reverse log-majorization statement complementing an earlier result in the paper, and as a route toward the preceding trace conjecture for $s\in[1,2]$. Its status is not resolved in the supplied text.