The immense memory requirements of state-of-the-art Mixture-of-Experts (MoE) models present a significant challenge for inference, often exceeding the capacity of a single accelerator. While offloading experts to host memory is a common solution, it introduces a severe I/O bottleneck over the PCIe bus, as the data-dependent nature of expert selection places these synchronous transfers directly on the critical path of execution, crippling performance. This paper argues that the I/O bottleneck can be overcome by trading a small amount of cheap, on-device computation to hide the immense cost of data movement. We present MoE-SpeQ, a new inference system built on a novel co-design of speculative execution and expert offloading. MoE-SpeQ employs a small, on-device draft model to predict the sequence of required experts for future tokens. This foresight enables a runtime orchestrator to prefetch these experts from host memory, effectively overlapping the expensive I/O with useful computation and hiding the latency from the critical path. To maximize performance, an adaptive governor, guided by an Amortization Roofline Model, dynamically tunes the speculation strategy to the underlying hardware. Our evaluation on memory-constrained devices shows that for the Phi-MoE model, MoE-SpeQ achieves at most 2.34x speedup over the state-of-the-art offloading framework. Our work establishes a new, principled approach for managing data-dependent memory access in resource-limited environments, making MoE inference more accessible on commodity hardware.