Recent advances in uncertainty quantification increasingly emphasise the distinction between aleatory and epistemic uncertainty in machine learning, motivating the need for more unified frameworks. However, despite much progress in producing reliable predictions, existing methods often lack rigorous guarantees when generalising beyond the training domain. We propose a conformalised imprecise inference framework for robust extrapolation, which is model-agnostic and augments predictive models with imprecision and distance awareness. The proposed approach yields imprecise predictions (probability boxes) that remain valid under distributional shift, maintaining coverage while adaptively expanding uncertainty in extrapolation regimes. Experiments on synthetic and benchmark datasets demonstrate improved robustness and reliable coverage compared to standard probabilistic approaches, particularly under limited data.