Jaegers' recursive monotone fixed-point conjecture

Let ${\mathcal{N}}$ denote Baire space. A partial ordering $\preceq$ on ${\mathcal{N}}$ is $\Pi^0_1$, every $\preceq$-chain has a least upper bound, and $f:{\mathcal{N}}\times{\mathcal{N}}\to{\mathcal{N}}$ has sections $f_\gamma:{\mathcal{N}}\to{\mathcal{N}}$. Jaegers' conjecture. There is such a partial ordering and function $f$ satisfying: (i) $f$ is recursive; (ii) every section $f_\gamma$ is monotone with respect to $\preceq$ and therefore has a fixed point; and (iii) no fixed point of $f_\gamma$ is $\operatorname{HYP}(\gamma)$. The conjecture proposes a Baire-space analogue of the preceding non-uniformization result, overcoming the compactness obstruction that prevents the corresponding function on Cantor space from being recursive; the necessary modification of the ordering is left implicit.