  The mutual information $I(A,B)$ of pairs of spatially separated regions
satisfies, for any $d$-dimensional CFT, a set of structural physical properties
such as positivity, monotonicity, clustering, or Poincar\'e invariance, among
others. If one imposes the extra requirement that $I(A,B)$ is extensive as a
function of its arguments (so that the tripartite information vanishes for any
set of regions, $I_3(A,B,C)\equiv 0$), a closed geometric formula involving
integrals over $\partial A$ and $\partial B$ can be obtained. We explore
whether this "Extensive Mutual Information" model (EMI), which in fact
describes a free fermion in $d=2$, may similarly correspond to an actual CFT in
general dimensions. Using the long-distance behavior of $I_{\rm
\scriptscriptstyle EMI}(A,B)$ we show that, if it did, it would necessarily
include a free fermion, but also that additional operators would have to be
present in the model. Remarkably, we find that $I_{\rm \scriptscriptstyle
EMI}(A,B)$ for two arbitrarily boosted spheres in general $d$ exactly matches
the result for the free fermion current conformal block
$G^d_{\Delta=(d-1),J=1}$. On the other hand, a detailed analysis of the
subleading contribution in the long-distance regime rules out the possibility
that the EMI formula represents the mutual information of any actual CFT or
even any limit of CFTs. These results make manifest the incompleteness of the
set of known constraints required to describe the space of allowed entropy
functions in QFT.
