  B\'ar\'any, Kalai, and Meshulam recently obtained a topological Tverberg-type
theorem for matroids, which guarantees multiple coincidences for continuous
maps from a matroid complex to d-dimensional Euclidean space, if the matroid
has sufficiently many disjoint bases. They make a conjecture on the
connectivity of k-fold deleted joins of a matroid with many disjoint bases,
which would yield a much tighter result - but we provide a counterexample
already for the case of k=2, where a tight Tverberg-type theorem would be a
topological Radon theorem for matroids. Nevertheless, we prove the topological
Radon theorem for the counterexample family of matroids by an index
calculation, despite the failure of the connectivity-based approach.
