In this paper by Diaconis and Zabell from 1982, Theorem 2.1 and the remark after essentially stated that

Given two probability measures $P$ and $Q$ on the same probability space $\Omega$. If $Q\ll P$ and their RN-derivative $\frac{dQ}{dP} \in L^\infty$ , then $Q$ can be obtained from $P$ by conditioning.

Their proof is very easy and the assumption $\frac{dQ}{dP} \in L^\infty$ is essential for it. However I was wondering if it is possible to relax the assumption $\frac{dQ}{dP} \in L^\infty$ or if this is a necessary condition, is it? I was looking at concrete cases of this problem, for example between $\text{Exp}(a)$ and $\text{Exp}(b)$.

I know very little about the literature on this topic, any suggestion is also appreciated.