The completeness theorem for exponential families says that if the natural parameter space contains a k-dimensional rectangle then (T1, ... , Tk) is complete. The proof can be found in 'Testing Statistical Hypotheses', Lehmann, 1986, page 142. I've included it below so you can read it easily.
What happens with the other implication of the theorem? Is there any complete statistic that doesn't contain a k-dimensional rectangle such that it can be used as a counterexample?