Bárány–Kalai–Meshulam conjecture on connectivity of deleted joins of matroids

Let $k\geq 1$ be an integer. For a matroid $M$ of rank $r\geq 1$, let $M^{*k}_\Delta$ denote its $k$-fold deleted join, and say that $M$ has disjoint bases when it contains bases that are pairwise disjoint.

Bárány–Kalai–Meshulam conjecture. There exists an integer $n_k\geq 1$ depending only on $k$ such that, whenever $M$ has at least $n_k$ disjoint bases, the complex $M^{*k}_\Delta$ is $(kr-1)$-dimensional and $(kr-2)$-connected.

The conjecture asks for a sharp connectivity statement for deleted joins of matroids, motivated by the connectivity-based approach to configuration space/test map schemes for Tverberg-type theorems. Its status is not resolved in the supplied source information.