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Mobile systems will have to support multiple AI-based applications, each leveraging heterogeneous data sources through DNN architectures collaboratively executed within the network. To minimize the cost of the AI inference task subject to requirements on latency, quality, and - crucially - reliability of the inference process, it is vital to optimize (i) the set of sensors/data sources and (ii) the DNN architecture, (iii) the network nodes executing sections of the DNN, and (iv) the resources to use. To this end, we leverage dynamic gated neural networks with branches, and propose a novel algorithmic strategy called Quantile-constrained Inference (QIC), based upon quantile-Constrained policy optimization. QIC makes joint, high-quality, swift decisions on all the above aspects of the system, with the aim to minimize inference energy cost. We remark that this is the first contribution connecting gated dynamic DNNs with infrastructure-level decision making. We evaluate QIC using a dynamic gated DNN with stems and branches for optimal sensor fusion and inference, trained on the RADIATE dataset offering Radar, LiDAR, and Camera data, and real-world wireless measurements. Our results confirm that QIC matches the optimum and outperforms its alternatives by over 80%.