We consider a Gaussian process model trained on few evaluations of an expensive-to-evaluate deterministic function and we study the problem of estimating a fixed excursion set of this function. We focus on conservative estimates as they allow control on false positives while minimizing false negatives. We introduce adaptive strategies that sequentially selects 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. We provide tractable formulae for the conservative criteria and we benchmark the method on random functions generated under the model assumptions in two and five dimensions. Finally the method is 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 a-posteriori indicators.