Agreement of pseudoétale and elementary finite-to-one topologies on diffuse manifolds

Let $\mathcal{M}$ be the subcategory of $C^n$-manifolds, where $n\in\mathbb{N}$ or $n=\infty$, consisting of all manifolds but only diffuse $C^n$-functions. Let the elementary finite-to-one topology be the topology generated by finite discrete local subsets. The topology-agreement conjecture. The pseudoétale and elementary finite-to-one topologies on $\mathcal{M}$ agree.

The elementary finite-to-one topology is the topology commonly used in the real case, while the source gives no proof of its agreement with the derived pseudoétale topology in this setting.