The exponents of the Ando-Hiai property for the Wasserstein mean

For positive definite matrices A and B and a weight t ∈ (0,1), the Wasserstein mean A ∘ₜ B minimizes (1-t)d(X,A)² + t\,d(X,B)² over positive definite X, where d is the Bures-Wasserstein distance. The mean of weight t has the Ando-Hiai property for an exponent $q > 0$ on m × m matrices if A ∘ₜ B ≤ I implies Aq ∘ₜ Bq ≤ I for all such A and B. Gan and Kim asked whether A² ∘ₜ B² ≤ I implies A ∘ₜ B ≤ I for every t, which is the property for the exponent $1/2$. They also conjectured that (Aᵖ ∘ₜ Bᵖ)1/p decreases in weak log-majorization as p decreases, where weak log-majorization compares the products of the k largest eigenvalues for each k. We determine the set of exponents with the property for each m and t. For q ≤ 1, the ratio of the largest eigenvalue of Aq ∘ₜ Bq to the qth power of that of A ∘ₜ B is at most c, where c = (1-t)2-2q + t2-2q for 1/2 ≤ q ≤ 1 and $c = 1$ for q ≤ 1/2, so every q ≤ 1/2 has the property. For m ≥ 3 and 1/2 ≤ q < 1 the ratio tends to c along an explicit family of m × m pairs. The exponent $1$ has the property trivially, every $q > 1$ fails and $c > 1$ for $1/2 < q < 1$ so the set is (0,1/2] ∪ \1\ for m ≥ 3, whereas a separate argument gives $(0, 1]$ for m ≤ 2. Hence Gan and Kim's implication holds for every t, whereas they stated it only for weights that depend on A and B. The conjecture fails at $k = 2$ for one 2 × 2 pair, and at $k = 1$ for m ≥ 3 because its case $k = 1$ is the property for every q ≤ 1.