No linear centering gap for octagons near an affine-regular hexagon

We study the unrestricted vertex index V(K), defined by minimizing the sum of gauges at the actual vertices of polygons enclosing an origin-symmetric planar convex body, and its restriction W(K) to origin-centered quadrilaterals. For linear deformations of four antipodal vertex pairs that yield genuine octagons and converge to an affine-regular hexagon H, we give a conditional candidate result: V and W have the same finite right derivative. Thus W(Kε) − V(Kε) = o(ε). For each fixed generic direction, both individual expansions have O(ε²) remainders and an explicit common first-order coefficient. The argument covers vertex splitting and insertion into opposite edge interiors. It localizes arbitrary actual minimizers, retains coalescing containment constraints, centers convex symmetric first-order costs, and clips a shrinking gauge plateau using the full objective. Uniform local Hausdorff control handles degenerate directions without asserting a uniform quadratic rate. The result is provisional and conditional on earlier candidate results giving the hexagon minimum and genuine-pentagon bound (H1), and the complete classification of its minimizing quadrilateral enclosures (H2). It does not assert exact equality V = W, centeredness of all minimizers, or an effective parameter cutoff.