Let $(X,\mathcal B)$ be a matroid on the ground set $X=(x_1,...,x_n)$ and with set of bases $\mathcal B$, and let $P\subset\Bbb R^n$ be its matroid base polytope (i.e. the convex hull of the characteristic vectors of the bases $B\in\mathcal B$). The circumcenter of $P$ is the unique point $p\in\mathrm{aff}(P)$ that has the same distance to all vertices of $P$.

Question: Does $P$ contain its circumcenter, perhaps even in its relative interior?

Here relative interior means the interior of $P$ considered as a subset of its affine hull.