In the comments on this answer, user Scortchi asks:
So iff there's a sufficient statistic of constant dimension, there's a conjugate prior?
As far as I know this didn't get a complete answer, so I'm asking it as a new question in the hope of finding out whether it's true. My question is the quote above; I give more details below.
This question can be seen as asking for a generalisation of the Pitman-Koopman-Darmois theorem, which states that if a family of distributions is such that the support does not depend on the parameters, and if the family has a sufficient statistic whose dimensionality doesn't change as the number of samples increases, then the family must be an exponential family.
We also have that if a family of distributions is such that the support does not depend on the parameters and the family admits a conjugate prior, then it must be an exponential family, which is a similar but different result.
However, as the example in the linked answer shows, if we relax the assumption that the support doesn't depend on the parameters, then it's possible for a distribution to have a conjugate prior without being an exponential family. The question is whether something similar happens if we relax the corresponding assumption in the Pitman-Koopman-Darmois theorem, and specifically, whether we end up with the same set of families of distributions in both cases.
In other words, the conjecture is that the Pitman-Koopman-Darmois theorem can be generalised into the following statement: "For an arbitrary family of distributions, whose support might depend on its parameters, the family has a conjugate prior if and only if it has a sufficient statistic whose dimensionality doesn't change as the number of samples increases." Is this statement true or false?