  Let $S$ be an operator system sitting in its C*-envelope
$C^*_{\mathrm{min}}(S)$. Starting with a pure state on $S$, let $F$ be the face
of state extensions to $C^*_{\mathrm{min}}(S)$. The dilation theorem of
Davidson-Kennedy shows that the GNS representations corresponding to some of
the extreme states of $F$ are boundary representations. We construct an
explicit example in which $F$ is an interval and only one of the two extreme
points yields a boundary representation.
