We consider the problem of estimating the set of all inputs that leads a system to some particular behavior. The system is modeled with an expensive-to-evaluate function, such as a computer experiment, and we are interested in its excursion set, i.e. the set of points where the function takes values above or below some prescribed threshold. The objective function is emulated with Gaussian Process (GP) models based on an initial design of experiments enriched with evaluation results at (batch-)sequentially determined input points. The GP model provides conservative estimates for the excursion set, which control false positives while minimizing false negatives. We introduce adaptive strategies that sequentially select new evaluations of the function by reducing the uncertainty on conservative estimates. Following the Stepwise Uncertainty Reduction approach we obtain new evaluations by minimizing adapted criteria. Tractable formulae for the conservative criteria are derived which allow more convenient optimization. The method is benchmarked on random functions generated under the model assumptions in two and five dimensions and applied to a reliability engineering test case. Overall, the proposed strategy of minimizing false negatives in conservative estimation achieves competitive performance both in terms of model-based and model-free indicators.