Non-reducibility of NIN to HBU for Baire class 2 functions

Let $\textup{\textsf{NIN}}$ denote the problem of finding a function witnessing non-injectivity, and let $\textup{\textsf{HBU}}$ denote the Heine–Borel theorem for covers of $[0,1]$. Restrict the covers in $\textup{\textsf{HBU}}$ to functions of Baire class $2$; also consider the full Lebesgue number lemma. Non-reducibility conjecture. The problem $\textup{\textsf{NIN}}$ does not $N$-reduce to $\textup{\textsf{HBU}}$ restricted to Baire class $2$ functions, nor to the full Lebesgue number lemma. The conjecture is motivated by the preceding argument that post-processing is necessary because a Turing machine cannot evaluate a third-order functional at a point due to type restrictions. The source gives no resolution.