  Let $K_d$ denote the class of all finite graphs and, for graphs $A\subseteq
B$, say $A \leq_d B$ if distances in $A$ are preserved in $B$; i.e. for $a, a'
\in A$ the length of the shortest path in $A$ from $a$ to $a'$ is the same as
the length of the shortest path in $B$ from $a$ to $a'$. In this situation
$(K_d, \leq_d)$ forms an amalgamation class and one can perform a Hrushovski
construction to obtain a generic of the class. One particular feature of the
class $(K_d, \leq_d)$ is that a closed superset of a finite set need not
include all minimal pairs obtained iteratively over that set but only enough
such pairs to resolve distances; we will say that such classes have
$\exists$-resolutions. Larry Moss conjectured the existence of graph $M$ which
was $(K_d, \leq_d)$-injective (for $A \leq_d B$ any isometric embedding of $A$
into $M$ extends to an isometric embedding of $B$ into $M$) but without finite
closures. We examine Moss's conjecture in the more general context of
amalgamation classes. In particular, we will show that the question is in some
sense more interesting in classes with $\exists$-resolutions and will give some
conditions under which the possibility of such structures is limited.
