In-Context Learning (ICL) has become a cornerstone of modern LLM deployment. However, existing ICL post-training methods have a critical blind spot: they excel at extracting patterns from demonstrations while often neglecting context authority, the ability to determine whether contextual information should govern the final answer. To benchmark this capability, we introduce FakeContextBench, which contains pseudoscientific claims across seven domains. Our evaluation of commercial and open-source models shows that large-scale pre-training alone is insufficient for reliable context-authority discrimination. Moreover, prevalent ICL fine-tuning methods can increase susceptibility to misleading context, reducing reality accuracy by up to 14.95 percentage points relative to the base model. To address this trade-off, we propose Jurisdiction In-Context Learning (J-ICL), a post-training framework that incorporates context validation into the training objective. Across four model backbones, J-ICL improves ICLEval by an average of 5.84 percentage points and reality accuracy by 9.20 points over the corresponding base models. It also raises the Reality Rate by an average of 18.09 points relative to MetaICL and Symbol Tuning. These results demonstrate that ICL capability and resistance to deceptive context can be improved together. The benchmark is available at https://github.com/peilin717/FakeContext-Bench.

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