Separation of choice principles in elementary topoi

Let $\mathbb{N}$ denote the natural numbers, let $\mathbf{2}=\{0,1\}\subseteq\mathbb{N}$, and let $\mathrm{AC}_{\mathbb{N}\text{-bounded}}$, $\mathrm{AC}_{\mathbb{N}\mathbb{N}}$, and $\mathrm{AC}_{\mathbb{N}\mathbf{2}}$ denote the choice principles defined in the paper. Let $\mathbb{R}_C$ and $\mathbb{R}_D$ be the Cauchy and Dedekind real objects, respectively.

Choice-separation conjecture.

1. There exists an elementary topos in which $\mathrm{AC}_{\mathbb{N}\text{-bounded}}$ holds but $\mathrm{AC}_{\mathbb{N}\mathbb{N}}$ does not.
2. There exists an elementary topos in which $\mathrm{AC}_{\mathbb{N}\mathbf{2}}$ holds but $\mathrm{AC}_{\mathbb{N}\text{-bounded}}$ does not.
3. There exists an elementary topos in which $\mathrm{AC}_{\mathbb{N}\mathbf{2}}$ holds but $\mathbb{R}_C\neq\mathbb{R}_D$.

These claims seek elementary-topos models separating bounded, natural-number-valued, and two-valued choice, and the third would imply the second. The supplied text provides no resolution.