Log-convexity of strip dimer counts on the king's graph

Let Z(k, m) be the number of perfect matchings of the k × m king board, the square grid with both diagonals, i.e. the number of its domicule tilings. In all 1453 cases we tested, on free strips of width m ≤ 26 and cylinders of circumference m ≤ 22, the sequence q ↦ Z(2q, m) is log-convex; the tests are exact or carry a rigorous rounding bound. The property is not structural. It fails on free strips of the triangular lattice, for every unequal pair of diagonal weights we tried (for nearly equal weights at m = 4 and 8 a Krein-signature argument for the left-right reflection predicts the onset exactly), and on wide strips for equal diagonal weights x = 3/2, 2, 3. Its limit, convexity of q ↦ log λ2q for the strip growth rates, would give a conditional lower bound on the entropy with ten correct digits. Along equal weights it is a competition between two modes of the transfer matrix whose rates cross, on our data, between x = 7/4 and x = 2. We collect the obstructions to the standard proof techniques and two exact reformulations.