Generalized eigenvalue log-majorization conjecture for geometric means

Let $A,B\geq0$, let $r,s\in\mathbb{R}$, and for $0\leq t\leq1$ define

$$A\sharp_{r,t}B=A^{r/2}(A^{-1/2}BA^{-1/2})^tA^{r/2}.$$

Let $\lambda(X)$ denote the vector of eigenvalues of $X$, arranged in decreasing order, and let $\prec_{\log}$ denote log-majorization.

Generalized geometric-mean conjecture. If $0\leq t\leq1$, $p\geq1$, and either $r,s\geq1$ or $r,s\leq0$, then

$$\lambda\left((A\sharp_{r,t}B)^p(A\sharp_{s,1-t}B)^p\right)\prec_{\log}\lambda\left(A^{p(r+s-1)}B^p\right).$$

The claim extends the paper’s established cases for restricted ranges of $p,r,s,t$. No resolution is supplied in the provided text.