This paper proposes a procedure for assessing the sensitivity of inferential conclusions for functionals of sparse high-dimensional models following model selection. The proposed procedure is called targeted undersmoothing. Functionals considered include dense functionals that may depend on many or all elements of the high-dimensional parameter vector. The sensitivity analysis is based on systematic enlargements of an initially selected model. By varying the enlargements, one can conduct sensitivity analysis about the strength of empirical conclusions to model selection mistakes. We illustrate the procedure's performance through simulation experiments and two empirical examples.