The Ando--Hiai property of the weighted spectral geometric mean

For positive definite matrices A and B and $0 0$ if AₜB≤ I implies AqₜBq≤ I. Seo, Wada and Yamazaki proved the property for $0 1$. At $t=1/2$, this bound equals $1$, so every exponent is settled at $t=1/2$. For t≠1/2, they asked whether the property holds for every q≤1. We determine the exponents that have the property for n× n matrices with n≥3. For these matrices, the property holds exactly for q≤min/(1-t),(1-t)/t\ and for $q=1$. Explicit 3×3 pairs violate it for every q strictly between these two values. For 2×2 matrices, one pair, certified in interval arithmetic, violates the property at $q=9/10$ for every t≤1/100, and the same pair with A and B exchanged violates it for every t≥99/100. In contrast, an argument in the hyperbolic plane, which the real 2×2 positive definite matrices of determinant $1$ form, proves the property for every q≤1 on the 2×2 pairs whose condition numbers satisfy (κ(B)-κ(A))(2t-1)≥0. For Hiai's quasi-geometric means t,p(A,B)=(AᵖₜBᵖ)1/p and $p<p'$, the inequality between the largest eigenvalues in the log-majorization t,plogt,p' holds for all pairs exactly when ₜ has the property for $p/p'$. Hence Hiai's necessary condition p≤ p' for this log-majorization is not sufficient. Two log-majorizations stated by Gan and Tam are false, namely t,plogt,p' for all 0<p≤ p' and AqₜBqlog(AₜB)q for all $0