For every order h such that the infinite sum of all terms 1/h(n) is finite, every K-trivial degree is h-jump-traceable. This fact motivated Cholak, Downey and Greenberg to ask whether this traceability property is actually equivalent to K-triviality, thereby giving the hoped for combinatorial characterization of lowness for Martin-Löf randomness. We show however that the K-trivial degrees are properly contained in those that are h-jump-traceable for every convergent order h.