Macbeath point inclusion conjecture

Let $K\subset\mathbb{R}^d$ be a compact convex set with non-empty interior, and let $p$ be its Macbeath point, namely the unique point in the interior of $K$ maximizing

$$\operatorname{vol}_d\left(K\cap(-K+2p)\right).$$

Macbeath point conjecture. The inclusion

$$K-p\subset -d(K-p)$$

holds. This is presented as an open problem related to volumetric Helly-type results. The uniqueness and existence of the Macbeath point are known from Macbeath's lemma, while the stated inclusion remains unresolved in the supplied text.