We prove that the statement “there is aksuch that for everyfthere is ak-bounded diagonally nonrecursive function relative tof” does not imply weak König’s lemma over${\rm{RC}}{{\rm{A}}_0} + {\rm{B\Sigma }}_2^0$. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that everyk-bounded diagonally nonrecursive function computes a 2-bounded diagonally nonrecursive function may fail in the absence of${\rm{I\Sigma }}_2^0$.