An exact counterexample to the generalized geometric-mean conjecture

For positive definite matrices let $A\#_{r,t}B=A^{r/2}(A^{-1/2}BA^{-1/2})^tA^{r/2}$. The note disproves the conjectured eigenvalue log-majorization $\lambda((A\#_{r,t}B)^p(A\#_{s,1-t}B)^p)\prec_{\log}\lambda(A^{p(r+s-1)}B^p)$ for $0\le t\le1$, $p\ge1$ and either $r,s\ge1$ or $r,s\le0$. It fails for $r=s=1$, $p=2$ and $t=1/8$, with $A=D^2$ and $B=DT^8D$, where $D=\operatorname{diag}(16,1/12,1)$ and $T$ is an explicit integer positive definite $3\times3$ matrix. Since $A^{-1/2}BA^{-1/2}=T^8$, the two means are exactly $G=DTD$ and $H=DT^7D$, so no fractional power is approximated. As $\lambda_{\max}(G^2H^2)=\|GH\|_2^2$ and $\lambda_{\max}(A^2B^2)=\|AB\|_2^2$, exact rational arithmetic, bounding one entry of $GH$ from below and $\|AB\|_F$ from above, shows that the largest eigenvalue on the left exceeds the one on the right. The example lies outside the interval $1/4\le t\le3/4$ where the inequality is known for $r=s=1$ and $p=2$.