Lemos–Soares singular-value log-majorization conjecture

Let $A$ and $B$ be positive semi-definite matrices, let $r,s\in\mathbb{R}$, and for $0\leq t\leq1$ define the generalized weighted geometric mean by

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

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

Lemos–Soares conjecture. If $A,B\geq0$, $r,s\in\mathbb{R}$, and $0\leq t\leq1$, then

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

This conjecture asks for a singular-value analogue of a previously known eigenvalue log-majorization inequality. The supplied status evidence indicates that it has been resolved, although the resolving result is not identified in the provided text.