Comparing the strength of diagonally non-recursive functions in the absence of Σ02 induction

We prove that the statement there is a k such that for every f there is a k-bounded diagonally non-recursive function relative to f does not imply weak K\ onig\u27s lemma over RCA0+BΣ02. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally non-recursive function computes a 2-bounded diagonally non-recursive function may fail in the absence of IΣ02