The rapid proliferation of intermittent renewable electricity generation demands a corresponding growth in grid-scale energy storage systems to enable grid decarbonization. To encourage investment in energy storage infrastructure, storage operators rely on forecasts of electricity prices along with uncertainty estimates to maximize profit while managing risk. However, well-calibrated uncertainty estimates can be difficult to obtain in high-capacity prediction models such as deep neural networks. Moreover, in high-dimensional settings, there may be many valid uncertainty estimates with varied performance profiles—i.e., not all uncertainty is equally valuable for downstream decision-making. To address this challenge, this paper develops an end-to-end framework for conditional robust optimization, with robustness and calibration guarantees provided by conformal prediction. We represent arbitrary convex uncertainty sets with partially input-convex neural networks, which are learned as part of our framework. We demonstrate the value of our approach for robust decision-making on a battery storage arbitrage application.