  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 $\mathrm{RCA}_0 + \mathrm{B}\Sigma^0_2$. 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
$\mathrm{I}\Sigma^0_2$.
