Chain-of-thought (CoT) reasoning and reasoning-tuned models such as DeepSeek-R1 are commonly assumed to reduce shallow heuristic biases by thinking carefully. We test this on position bias in multiple-choice QA and find a different story: within any reasoning-capable model, per-question position bias scales with the length of the reasoning trajectory. Across thirteen reasoning-mode configurations (two R1-distilled 7-8B models, two base models prompted with CoT, and DeepSeek-R1 at 671B) on MMLU, ARC-Challenge, and GPQA, twelve show a positive partial correlation between trajectory length and Position Bias Score (PBS) after controlling for accuracy, ranging from 0.11 to 0.41 (all p < 0.05). All twelve open-weight reasoning-mode configurations show monotonically increasing PBS across length quartiles. A truncation intervention provides causal evidence: continuations resumed from later points in the trajectory are increasingly likely to shift toward position-preferred options (16% to 32% for R1-Qwen-7B across absolute-position buckets). At 671B, aggregate PBS collapses to 0.019, but the length effect still manifests in the longest quartile (PBS = 0.071), suggesting that accuracy gates the expression of length-driven bias rather than eliminating the underlying mechanism. We additionally find that direct-answer position bias is a distinct phenomenon with a different footprint (strong in Llama-Instruct-direct, weak in Qwen-Instruct-direct, and uncorrelated with trajectory length): CoT reasoning replaces this baseline bias with length-accumulated bias. Our results argue that reasoning-capable models should not be treated as order-robust by default in MCQ evaluation pipelines, and offer a diagnostic toolkit (PBS, commitment change point, effective switching, truncation probes) for auditing position bias in reasoning models.