Weak Rota basis conjecture for scrambled systems in odd dimensions

Let $S_1,\ldots,S_n$ be a scrambled system of bases of $\mathbb{R}^n$: each $S_i$ has size $n$, and $\bigcup S_i$ is the multiset union of $n$ bases. An independent transversal contains one element from each $S_i$ and is linearly independent. Weak Rota basis conjecture. A scrambled system of bases of $\mathbb{R}^n$ contains $n-1$ disjoint independent transversals. The paper proposes this as an analogue of its odd-dimensional result; the supplied text gives no resolution of the claim.