Macbeath point inclusion conjecture

Prove that for every compact convex set K ⊂ R^d with non-empty interior, if p is the Macbeath point of K (the unique maximizer of x ↦ vol(K ∩ (−K + 2x))), then the inclusion K − p ⊂ −d(K − p) holds.

Background

Macbeath (1952) showed that for a compact convex set K with non-empty interior, the function f(x) = vol(K ∩ (−K + 2x)) attains its maximum at a unique interior point, here termed the Macbeath point. Centers "deep" inside convex bodies (e.g., John center, barycenter, Santaló point) often play a crucial role in quantitative Carathéodory- and Helly-type results.

The conjectured inclusion characterizes a strong symmetry/dilation property of K around its Macbeath point and, if true, could have implications for volumetric Helly-type results and coarse approximation of convex bodies.