Regular polygons as potential counterexamples to the ball-maximizer inequality

Ascertain whether regular polygons P ⊂ ℝ^2 are counterexamples to the inequality vol(C_δ P) ≤ vol(C_δ B_P) for some δ ∈ (0,1), where B_P denotes the Euclidean disk with the same area as P; equivalently, determine whether there exists δ ∈ (0,1) such that vol(C_δ P) > vol(C_δ B_P).

Background

The authors show, via a family of smooth symmetric perturbations of the disk, that for every δ ∈ (0,1) there exist convex bodies K ⊂ ℝ^2 with vol(C_δ K) > vol(C_δ B_K), giving a negative answer to a natural maximization question in the plane. They note, however, that previous direct computations attempting to use regular polygons as such counterexamples for δ near 1 failed, likely due to complicated oscillatory behavior in the second-order analysis.

This leaves open whether regular polygons specifically (as a prominent class of planar convex bodies) can serve as counterexamples to the inequality comparing C_δ K with C_δ of the equal-area disk, for some value(s) of δ.