Cauchy-completeness separation conjecture for elementary topoi

Let $\mathbb{R}_C$ and $\mathbb{R}_E$ denote the Cauchy and Euclidean real objects, respectively, and let $\mathrm{AC}_{\mathbb{N}\mathbf{2}}$ be the two-valued choice principle defined in the paper.

Cauchy-completeness separation conjecture.

1. There exists an elementary topos in which $\mathbb{R}_C\neq\mathbb{R}_E$; equivalently, there exists an elementary topos in which $\mathbb{R}_C$ is not Cauchy complete.
2. There exists an elementary topos in which $\mathrm{AC}_{\mathbb{N}\mathbf{2}}$ holds but $\mathbb{R}_C\neq\mathbb{R}_E$.

The second assertion is stated to imply both the first assertion and the corresponding earlier conjecture involving $\mathbb{R}_C\neq\mathbb{R}_D$. The supplied text gives no resolution.