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Edge-assisted Augmented Reality (AR) which offloads computeintensive Deep Neural Network (DNN)-based AR tasks to edge servers faces an important design challenge: how to pick the DNN model out of many choices proposed for each AR task for offloading. For each AR task, e.g., depth estimation, many DNN-based models have been proposed over time that vary in accuracy and complexity. In general, more accurate models are also more complex; they are larger and have longer inference time. Thus choosing a larger model in offloading can provide higher accuracy for the offloaded frames but also incur longer turnaround time, during which the AR app has to reuse the estimation result from the last offloaded frame, which can lead to lower average accuracy. In this paper, we experimentally study this design tradeoff using depth estimation as a case study. We design optimal offloading schedule and further consider the impact of numerous factors such as on-device fast tracking, frame downsizing and available network bandwidth. Our results show that for edge-assisted monocular depth estimation, with proper frame downsizing and fast tracking, compared to small models, the improved accuracy of large models can offset its longer turnaround time to provide higher average estimation accuracy across frames under both LTE and 5G mmWave.