Comparing the strength of diagonally non-recursive functions in the absence of ${\mathrm{\Sigma }}_2^0$ 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’s lemma over ${\mathrm{R}\mathrm{C}\mathrm{A}}_0+\mathrm{B}{\mathrm{\Sigma }}_2^0$. 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 $\mathrm{I}{\mathrm{\Sigma }}_2^0$.