Certain combinatorial optimization problems with binary representation require the candidate solutions to satisfy a balancedness constraint (e.g., being composed of the same number of 0s and 1s). A common strategy when using Genetic Algorithms (GA) to solve these problems is to use crossover and mutation operators that preserve balancedness in the offspring. However, it has been observed that the reduction of the search space size granted by such tailored variation operators does not usually translate to a substantial improvement of the GA performance. There is still no clear explanation of this phenomenon, although it is suspected that a balanced representation might yield a more irregular fitness landscape, where it could be more difficult for GA to converge to a global optimum. In this paper, we investigate this issue by adding a local search step to a GA with balanced operators, and use it to evolve highly nonlinear balanced Boolean functions. We organize our experiments around two research questions, namely if local search (1) improves the convergence speed of GA, and (2) decreases the population diversity. Surprisingly, while our results answer affirmatively the first question, they also show that adding local search actually increases the diversity among the individuals. We link these findings to some recent results on fitness landscape analysis for problems on Boolean functions.