The high energy physics unfolding problem is an important statistical inverse problem in data analysis at the Large Hadron Collider (LHC) at CERN. The goal of unfolding is to make nonparametric inferences about a particle spectrum from measurements smeared by the finite resolution of the particle detectors. Previous unfolding methods use ad hoc discretization and regularization, resulting in confidence intervals that can have significantly lower coverage than their nominal level. Instead of regularizing using a roughness penalty or stopping iterative methods early, we impose physically motivated shape constraints: positivity, monotonicity, and convexity. We quantify the uncertainty by constructing a nonparametric confidence set for the true spectrum, consisting of all those spectra that satisfy the shape constraints and that predict the observations within an appropriately calibrated level of fit. Projecting that set produces simultaneous confidence intervals for all functionals of the spectrum, including averages within bins. The confidence intervals have guaranteed conservative frequentist finite-sample coverage in the important and challenging class of unfolding problems for steeply falling particle spectra. We demonstrate the method using simulations that mimic unfolding the inclusive jet transverse momentum spectrum at the LHC. The shape-constrained intervals provide usefully tight conservative inferences, while the conventional methods suffer from severe undercoverage.