Conjecture on reductions is disproved

In a recent paper on the Secretary Problem, Abdolazimi et al. disproved our conjecture on reductions of matroids. The conjecture suggested that any matroid has a reduction to a partition matroid with covering number at most twice that of the original matroid. Though the statement was shown to hold for various matroid classes, it turned out to be false, e.g. for complete binary matroids of rank at least 17.