Hyperarithmetical pseudojump inversion conjecture

Let $e\in\mathbb{N}$ and let $A$ be a real such that

$$\mathcal{O}\mathrel{\leq_{\mathrm{T}}}A.$$

For a real $B$, define the $e$-th pseudo-hyperjump by $\operatorname{HJ}_e(B)=B\oplus V_e^B$, where $V_e^B$ is the $e$-th member of a uniformly effective enumeration of the $\Pi^{1,B}_1$ predicates. Hyperarithmetical pseudojump inversion conjecture. There exists a real $B$ such that

$$A\mathrel{\equiv_{\mathrm{T}}}\operatorname{HJ}_e(B)\mathrel{\equiv_{\mathrm{T}}}B\oplus\mathcal{O}.$$

This is the hyperarithmetical analogue of the Pseudojump Inversion Theorem, and the source explicitly states that it remains open.