Let $S$ be a scoring rule for probability functions. Define

$EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$.

Say that $S$ is striclty proper if and only if $P$ always minimises $EXP_{S}(Q|P)$ as a function of $Q$. Define

$D_{S}(P, Q) = EXP_{S}(Q|P) - EXP_{S}(P|P)$.

If $S$ is the logarithmic scoring rule defined by $S(P, w) = -ln(P(w))$, then $D_{S}(P, Q)$ is just the Kullback-Leibler divergence between $P$ and $Q$, or equivalently, the inverse Kullback-Leibler divergence between $Q$ and $P$. Note that the inverse Kullback-Leibler divergence is an $f$-divergence.

My question is this: *is there any other strictly proper scoring rule $S$ such that $D_{S}(P, Q)$ is equal to $F(Q, P)$ for some $f$-divergence $F$?*

I think that $D_{S}(P, Q)$ is always a Bregman divergence, and Amari proved that the only $f$-divergence that is also a Bregman divergence is the Kullback-Leibler divergence (on the space of probability functions). Is this enough to imply that there are no other strictly proper scoring rules with this property?