Counterparts of effective-topos local operators in realizability toposes

Let $\a0\boldsymbol{\mathsf{Eff}}$ be the effective topos, and let $\a0\mathsf{RT}(\mathcal{K}_2)$, $\a0\mathsf{RT}(\mathcal{K}_2^\mathrm{REC},\mathcal{K}_2)$, and $(\mathsf{Eff}\downarrow\Delta)$ be the realizability toposes described in the paper. A local operator is a modality on a topos; the conjecture concerns some of the local operators in $\a0\mathsf{Eff}$ considered by Lee and Van Oosten.

Counterpart conjecture. Some of these local operators have counterparts in the toposes

$$\mathsf{RT}(\mathcal{K}_2),\qquad \mathsf{RT}(\mathcal{K}_2^\mathrm{REC},\mathcal{K}_2),\qquad (\mathsf{Eff}\downarrow\Delta).$$

Van Oosten had already shown that the original Lifschitz realizability model has a counterpart in $\a0\mathsf{RT}(\mathcal{K}_2)$, and that a related result holds for $\a0\mathbf{q}$-realizability. The conjecture proposes analogous counterparts for some of the local operators, but the source does not establish which operators have them or prove the claim.