Conformance checking (CC) techniques of the process mining field gauge the conformance of the events constituting a case with respect to a business process model. Online conformance checking (OCC) techniques assess such conformance on streaming events. The realistic assumption of having a finite memory for storing the streaming events has largely not been considered by the OCC techniques. We propose a novel approach to reduce the memory consumption in prefix-alignment-based OCC techniques that ensures at the same time a minimal compromise in the conformance estimation quality. In alignment-based CC, states are used to reflect the results of the intermediate alignment steps. Our proposed approach bounds the number of maximum states in a prefix-alignment to be retained by any case in memory. It forgets the states in excess to the defined limit and retains its meaningful summary. Computing prefix-alignments in the future is then resumed for such cases from the current position contained in this summary. We highlight the superiority of our proposed approach compared to a state of the art OCC prefix-alignment-based approach with an infinite memory through experiments using real-life event data under a streaming setting. Our approach substantially reduces memory consumption while introducing at times a minor accuracy drop.