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Arguments in argumentation systems cannot always be considered as standalone entities, requiring the consideration of the pieces of evidence they rely on. This evidence might have to be retrieved from external sources such as databases or the web, and each attempt to retrieve a piece of evidence comes with an associated cost. Moreover, a piece of evidence may be available in a given scenario but not in others, and this is not known beforehand. As a result, the collection of active arguments (whose entire set of evidence is available) that can be used by the argumentation machinery of the system may vary from one scenario to another. In this work, we consider an Abstract Argumentation Framework with Fallible Evidence that accounts for these issues, and propose a heuristic measure used as part of the acceptability calculus (specifically, for building pruned dialectical trees) with the aim of minimizing the evidence retrieval cost of the arguments involved in the reasoning process. We provide an algorithmic solution that is empirically tested against two baselines and formally show the correctness of our approach.