Let $\mathscr C$ be a [locally strongly presentable category](https://arxiv.org/abs/0810.2578) (i.e. a cocomplete category that is the free cocompletion under sifted colimits of some small category), and let $X \in \mathscr C$. Is it the case that the coslice $X/\mathscr  C$ is also locally strongly presentable?

The analogous property holds for local $\kappa$-presentability (Proposition 1.57 of Adámek–Rosicky's *Locally presentable and accessible categories*), but does not generally hold for the general notion of $\mathbb D$-accessibility for a [sound limit doctrine](https://www.sciencedirect.com/science/article/pii/S0022404902001263) $\mathbb D$ (Remark 83 of Centazzo's [*Generalised algebraic models*](https://pul.uclouvain.be/resources/titles/29303100658430/extras/70957_sciences_centazzov3_1002422.pdf)). The proof in AR for traditional accessibility makes use of syntactic arguments (for instance Example 1.41), which do not have obvious analogues for other notions of accessibility, so the conceptual reason the coslice result holds for $\kappa$-presentability is unclear to me.