The Wikipedia page for conjugate priors lists several examples. Of the ones I'm immediately familiar with, all are exponential families. This leads me to wonder whether all families distributions that admit conjugate priors are exponential families.
More explicitly: suppose that $D$ is a family of distributions over $X$, parameterised by $\Theta$, that is, a probability measure $p(x;\theta)$ for each $\theta\in\Theta$. Suppose $D$ admits a conjugate prior. Does it follow that the family $D$ is an exponential family? If not, what is a simple counterexample, i.e. a family of distributions that admits a conjugate prior but is not an exponential family?
Uniform model with Pareto prior