For the completeness of links between MO questions, I posted an answer to a related question showing that Scott's formula $((¬¬p⇒ p) ⇒ (p∨¬p)) ⇒ (¬¬p∨¬p)$ does not hold in the Effective topos $\mathsf{RT}(\mathcal{K}_1)$ but does hold in the Kleene-Vesley topos $\mathsf{RT}(\mathcal{K}_2,\mathcal{K}_2^{\mathrm{rec}})$; if I'm not mistaken, the proof also applies to $\mathsf{RT}(\mathcal{K}_2^{\mathrm{rec}})$. So it also provides an answer to the present question.
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