Pseudo-Hyperjump Inversion Fails for Turing Degrees

Jananthan and Simpson asked in Conjecture~4.5 whether every pseudo-hyperjump $\operatorname{HJ}_e(X)=X\oplus V_e^X$, with $V_e^X$ uniformly $Π^1_1(X)$, admits Turing-degree inversion above Kleene's $\mathcal{O}$. We settle their conjecture in the negative. We construct one index $e_*$ such that $X<_