Relative Scott adjunction conjecture for Grothendieck topoi

Let $\mathcal{E}$ be a Grothendieck topos, and let $\omega \leq \lambda \leq \kappa$. Relative Scott adjunction conjecture. There is a $2$-adjunction

$$\mathsf{S}_{\mathcal{E}}: \mathcal{E}\text{-Acc}_{\omega} \leftrightarrows \operatorname{Topoi}_{/\mathcal{E}}: \mathsf{pt}_{\mathcal{E}}.$$

Moreover, if $\mathcal{E}$ is a $\kappa$-topos, the relative version of the $\lambda$-Scott adjunction holds for every $\omega \leq \lambda \leq \kappa$. This is proposed as a relative version of the Scott adjunction over a Grothendieck topos; the source frames it as a possibility and gives no evidence of a proof or resolution.