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Proximity detection is to determine whether an IoT receiver is within a certain distance from a signal transmitter. Due to its low cost and high popularity, Bluetooth low energy (BLE) has been used to detect proximity based on the received signal strength indicator (RSSI). To address the fact that RSSI can be markedly influenced by device carriage states, previous works have incorporated RSSI with inertial measurement unit (IMU) using deep learning. However, they have not sufficiently accounted for the impact of multipath. Furthermore, due to the special setup, the IMU data collected in the training process may be biased, which hampers the system's robustness and generalizability. This issue has not been studied before. We propose PRID, an IMU-assisted BLE proximity detection approach robust against RSSI fluctuation and IMU data bias. PRID histogramizes RSSI to extract multipath features and uses carriage state regularization to mitigate overfitting due to IMU data bias. We further propose PRID-lite based on a binarized neural network to substantially cut memory requirements for resource-constrained devices. We have conducted extensive experiments under different multipath environments, data bias levels, and a crowdsourced dataset. Our results show that PRID significantly reduces false detection cases compared with the existing arts (by over 50%). PRID-lite further reduces over 90% PRID model size and extends 60% battery life, with a minor compromise in accuracy (7%).