My question is in the title, but here is a more detailed formulation:

Let Top be the category of all topological spaces and continuous maps, and let CGTop be the subcategory of compactly generated spaces, as defined in Strickland's notes. We have a functor $k: Top \rightarrow CGTop$ that replaces the topology on a space by the associated compactly generated topology.

If X is a simplicial topological space (a simplicial object in the category of all topological spaces) is the identity map $|kX|\rightarrow k|X|$ a homeomorphism?

Here $|-|$ denotes geometric realization. I'm actually most interested in the "thick" realization (where the defining equivalence relation uses only the face maps), but the question makes sense for both the thick and thin versions.

It follows from the results in the above notes that $|kX|$ is compactly generated (being a quotient of a disjoint union of compactly generated spaces), and that the identity map $|kX|\rightarrow k|X|$ is continuous.

Here is a more general question: if $X$ is a topological space and $\sim$ is an equivalence relation on $X$, then $(kX)/{\sim}$ is compactly generated and the identity map $(kX)/{\sim} \to k(X/{\sim})$ is continuous. Is this map always a homeomorphism?