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The proliferation of machine learning services in the last few years has raised data privacy concerns. Homomorphic encryption (HE) enables inference using encrypted data but it incurs 100x-10,000x memory and runtime overheads. Secure deep neural network (DNN) inference using HE is currently limited by computing and memory resources, with frameworks requiring hundreds of gigabytes of DRAM to evaluate small models. To overcome these limitations, in this paper we explore the feasibility of leveraging hybrid memory systems comprised of DRAM and persistent memory. In particular, we explore the recently-released Intel Optane PMem technology and the Intel HE-Transformer nGraph to run large neural networks such as MobileNetV2 (in its largest variant) and ResNet-50 for the first time in the literature. We present an in-depth analysis of the efficiency of the executions with different hardware and software configurations. Our results conclude that DNN inference using HE incurs on friendly access patterns for this memory configuration, yielding efficient executions., We would like to thank Jesus Labarta from BSC and Steve Scargall from Intel for their insightful and productive comments., Peer Reviewed, Postprint (author's final draft)