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With rapid evolution of mobile core network (MCN) architectures, large-scale control-plane traffic (CPT) traces are critical to studying MCN design and performance optimization by the R&D community. The prior-art control-plane traffic generator SMM heavily relies on domain knowledge which requires re-design as the domain evolves. In this work, we study the feasibility of developing a high-fidelity MCN control plane traffic generator by leveraging generative ML models. We identify key challenges in synthesizing high-fidelity CPT including generic (to data-plane) requirements such as multimodality feature relationships and unique requirements such as stateful semantics and long-term (time-of-day) data variations. We show state-of-the-art, generative adversarial network (GAN)-based approaches shown to work well for data-plane traffic cannot meet these fidelity requirements of CPT, and develop a transformer-based model, CPT-GPT, that accurately captures complex dependencies among the samples in each traffic stream (control events by the same UE) without the need for GAN. Our evaluation of CPT-GPT on a large-scale control-plane traffic trace shows that (1) it does not rely on domain knowledge yet synthesizes control-plane traffic with comparable fidelity as SMM; (2) compared to the prior-art GAN-based approach, it reduces the fraction of streams that violate stateful semantics by two orders of magnitude, the max y-distance of sojourn time distributions of streams by 16.0%, and the transfer learning time in deriving new hourly models by 3.36×.