It is well-known that $f$-divergences defined on $\mathcal P(\mathcal X)$ where $\mathcal X$ is a measure space with $\sigma$-algebra $\mathcal B$ satisfy the property of information monotonicity:

For every $f$-divergence and every stochastic kernel \begin{equation*} \begin{split} K: \mathcal{X} \times \mathcal{B} \mapsto[0,1] \\ D_{f}(K P, K Q) \leq D_{f}(P, Q) \end{split} \end{equation*} Ref:

Thm 14. Liese, Vadja (2006). On Divergences and Informations in Statistics and Information Theory.

Conversely, if a divergence can be written as an integral of some function, and if it satisfies information monotonicity, then Amari showed that it must be an $f$-divergence. Ref: *Appendix A. Amari (2009). Alpha-Divergence Is Unique, Belonging to Both -Divergence and Bregman Divergence Classes*

This led Amari to conjecture, in the same paper that: When a divergence $D$ satisfies information monotonicity, it must be a function of an $f$ -divergence.

I haven't found any follow-ups to this conjecture in the literature. Does anyone know a reference for a proof or a counterexample?

Information Geometry and its Applicationsbook? $\endgroup$