A rational counterexample to the geometric-mean norm conjecture

For positive definite matrices let
$A\#_tB=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$. The note disproves the norm
inequality
$\|\sum_i(A_i^s\#_tB_i^s)^r\|\le\|(A^{(1-t)srp/2}B^{tsrp}A^{(1-t)srp/2})^{1/p}\|$,
with $A=\sum_iA_i$ and $B=\sum_iB_i$, conjectured for every unitarily
invariant norm, $0\le t\le1$ and $p,r,s>0$ with $sr\ge1$. It fails for two
rational positive definite $2\times2$ summands, the operator norm,
$s=t=1/2$ and $r=2$, for every $p>0$. The inputs are $A_1=C^2$, $A_2=E^2$,
$B_1=SE^2S$ and $B_2=SC^2S$, with diagonal $C,E>0$ satisfying
$C^2+E^2=I_2$ and a rational symmetric involution $S$, so $A=B=I_2$ and the
right side equals $1$. A closed formula for the geometric mean of $2\times2$
positive definite matrices, from the Cayley–Hamilton identity, makes the left
matrix rational, and it has the eigenvalue $1351000/1350907>1$.