  We prove that in a certain class of conformal data on an asymptotically
cylindrical manifold, if the conformally decomposed Einstein constraint
equations do not admit a solution, then one can always find a nontrivial
solution to the limit equation first explored by Dahl, Gicquaud, and Humbert in
[DGH11]. We also give an example of a Ricci curvature condition on the manifold
which precludes the existence of a solution to this limit equation, showing
that such a limit criterion can be a useful tool for studying the Einstein
constraint equations on manifolds with asymptotically cylindrical ends.
