In relation to Kostant's problem for simple highest weight modules over the general linear Lie algebra, we prove a persistence result for Kostant negative consecutive patterns. Inspired by it, we introduce the notion of a Kostant cuspidal permutation as a minimal Kostant negative consecutive pattern. It is shown that Kostant cuspidality is an invariant of a Kazhdan-Lusztig left cell. We describe four infinite families of Kostant cuspidal involutions, including a complete classification of Kostant cuspidal fully commutative involutions. In particular, we show that the number of new Kostant cuspidal elements can be arbitrarily large, when the rank grows. This provides some potential explanation why Kostant's problem is hard.