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

For positive definite matrices $A$ and $B$ and $0<t<1$, the weighted spectral geometric mean is $A\nat_tB=(A^{-1}\#B)^tA(A^{-1}\#B)^t$, where $\#$ denotes the geometric mean. The mean has the Ando--Hiai property for an exponent $q>0$ if $A\nat_tB\le I$ implies $A^q\nat_tB^q\le I$. Seo, Wada and Yamazaki proved the property for $0<q\le\min\{t/(1-t),(1-t)/t\}$ and disproved it for every $q>1$. At $t=1/2$, this bound equals $1$, so every exponent is settled at $t=1/2$. For $t\neq1/2$, they asked whether the property holds for every $q\le1$. We determine the exponents that have the property for $n\times n$ matrices with $n\ge3$. For these matrices, the property holds exactly for $q\le\min\{t/(1-t),(1-t)/t\}$ and for $q=1$. Explicit $3\times3$ pairs violate it for every $q$ strictly between these two values. For $2\times2$ matrices, one pair, certified in interval arithmetic, violates the property at $q=9/10$ for every $t\le1/100$, and the same pair with $A$ and $B$ exchanged violates it for every $t\ge99/100$. In contrast, an argument in the hyperbolic plane, which the real $2\times2$ positive definite matrices of determinant $1$ form, proves the property for every $q\le1$ on the $2\times2$ pairs whose condition numbers satisfy $(\kappa(B)-\kappa(A))(2t-1)\ge0$. For Hiai's quasi-geometric means $\SG_{t,p}(A,B)=(A^p\nat_tB^p)^{1/p}$ and $p<p'$, the inequality between the largest eigenvalues in the log-majorization $\SG_{t,p}\prec_{\log}\SG_{t,p'}$ holds for all pairs exactly when $\nat_t$ has the property for $p/p'$. Hence Hiai's necessary condition $p\le p'$ for this log-majorization is not sufficient. Two log-majorizations stated by Gan and Tam are false, namely $\SG_{t,p}\prec_{\log}\SG_{t,p'}$ for all $0<p\le p'$ and $A^q\nat_tB^q\prec_{\log}(A\nat_tB)^q$ for all $0<q\le1$.