Embeddings between partial combinatory algebras at successive Turing jumps

Let $X\subseteq\omega$. Write $\mathcal{K}_1^X$, $\mathcal{K}_2^X$, and $\mathcal{B}^X$ for the partial combinatory algebras defined in the paper, and write $X'$ for the Turing jump of $X$. The successive-jump embedding conjecture. For every $X\subseteq\omega$, there exist embeddings

$$\mathcal{K}_2^X\hookrightarrow\mathcal{K}_1^{X'}$$

and

$$\mathcal{B}^X\hookrightarrow\mathcal{K}_2^{X'}.$$

The preceding results characterize embeddings of $\mathcal{B}^X$ into $\mathcal{K}_1^Y$ in terms of the condition $X”\leqslant_TY$, and the conjecture proposes analogous embeddings for the other partial combinatory algebras. The supplied text gives no resolution of this conjecture.