Is Whyburn's theorem on irreducible maps optimal?

Whyburn's theorem (1939) says that a continuous surjection $q:Y\to X$, with $Y$ and $X$ compact and metrisable topological spaces, is irreducible if and only if the set of singleton fibres is dense in $Y$. Here irreducible means that there is no proper closed subset of $Y$ that $q$ maps surjectively onto $X$. I am not sure if it is part of the original theorem, but one can show that the set of singleton fibres is a $G_{\delta}$ in $Y$.

My question is, can the map $q$ be irreducible and the set of non-singleton fibres be dense in $Y$? I presume that the answer is yes, but I can't think of an example.

Let $C$ be the middle-thirds Cantor set and consider the equivalence relation $\sim$ on $C$ defined by $x\sim y$ iff there is an interval removed during the construction of the Cantor set whose endpoints are $x$ and $y$.

The quotient map $\pi\colon C\to C/\sim\cong[0,1]$ is an example of an irreducible map which has a dense set of non-singleton fibers.