Before revision:  For dense graphs we give lower bounds on the quantum query complexity of } \Omega(n ). Our upper bounds follow by combining a lemma on the structure of near-minimum cuts due to Rubinstein, Schramm and Weinberg (ITCS 2018) with a quantum sparsification algorithm) for \tau > 1 and \Omega(\tau n) for any 1 \leq \tau \leq n.
After revision:  For dense graphs we give lower bounds on the quantum query complexity of } \Omega(n ^{3/2) for \tau > 1 and \Omega(\tau n) for any 1 \leq \tau \leq n.