Decomposable concordances induce closed immersions

We show that a decomposable Lagrangian concordance between two Legendrian links must induce a closed immersion between the sheaf moduli of its ends. As the main application of this criterion, we prove a conjecture of Cornwell-Ng-Sivek, showing that their Lagrangian concordance is not decomposable. This establishes the first known example of a regular Lagrangian concordance between Legendrian knots that is not decomposable. In addition, we show that even the simpler Lagrangian concordance, satelliting via the Reeb 2-copy, is not decomposable.