Catenarity and epimorphisms of rings

Let $$R$$ be a commutative ring. The following are well-known:

1. If $$R$$ is catenary and $$\mathfrak{a}\subseteq R$$ is an ideal, then $$R/\mathfrak{a}$$ is catenary.
2. If $$R$$ is catenary and $$S\subseteq R$$ is a subset, then $$S^{-1}R$$ is catenary.

This means that catenarity is preserved along two special kinds of epimorphisms of commutative rings. In view of this question (and its answers) one may wonder:

Suppose that $$R$$ is catenary, and let $$f\colon R\twoheadrightarrow R'$$ be an epimorphism of commutative rings. Is $$R'$$ catenary?