A weak version of Rota's basis conjecture for odd dimensions

The Alon-Tarsi Latin square conjecture is extended to odd dimensions by stating it for reduced Latin squares (Latin squares having the identity permutation as their first row and first column). A modified version of Onn's colorful determinantal identity is used to show how the validity of this conjecture implies a weak version of Rota's basis conjecture for odd dimensions, namely that a set of $n$ bases in $\mathbb{R}^n$ has $n-1$ disjoint independent transversals.