Is there a cubic polynomial $f(x)$ with real coefficients such that $f$ is monotonic and $f(x)=f^{-1}(x)$ has more than $3$ real roots?

Does there exist a cubic polynomial $f(x)$ with real coefficients such that $f$ is monotonic (when regarded as a function from $\mathbb{R}$ to $\mathbb{R}$), and such that the equation $f(x)=f^{-1}(x)$ has more than $3$ real roots?

I couldn't find such an $f$, but I couldn't prove that no such $f$ exists.

It's clear that any such $f$ must be monotonically decreasing.

from Hot Weekly Questions - Mathematics Stack Exchange