Scrambled Rota conjecture
Let [math]M=(S,r)[/math] be a loopless matroid of rank k whose ground set can be partitioned into k bases. Is it true that no matter how we partition S into sets of size k, the partition will have k-1 disjoint transversals that are bases?
This was proposed by Aharoni and Kotlar . They give a simple example showing that k-1 cannot be replaced by k in the conjecture. A somewhat similar problem is the special case of Upper bound on common independent set cover when one of the matroids is a partition matroid. See also the survey of Aharoni, Charbit and Howard  that contains several related questions.