Speculative decoding accelerates autoregressive automatic speech recognition (ASR), but the acceptance length of alignment-aware drafters can saturate as the draft horizon increases. We propose a progress-aware speculative drafter that recurrently propagates an acoustic progress state across draft steps and feeds it back into audio cross-attention to guide token generation. We jointly train the drafter and progress predictor over variable draft horizons. On five ASR test sets, our method achieves lossless, macro-averaged end-to-end speedups of 1.657x and 1.227x over target-only autoregressive decoding with Qwen3-ASR-0.6B and Qwen3-ASR-1.7B, respectively. Relative to AnchorDraft, our method improves the macro-averaged speedup by 34.3% and 9.0%, respectively. Horizon sweeps show continued growth in acceptance length beyond the baselines' saturation. Code is available at https://github.com/yuanyuanjia71-spec/ProgDraft.

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