Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts

We establish a Gromov compactness theorem for strip shrinking in pseudoholomorphic quilts when composition of Lagrangian correspondences is immersed. In particular, we show that figure eight bubbling occurs in the limit, argue that this is a codimension-$0$ effect, and predict its algebraic consequences -- geometric composition extends to a curved $A_\infty$-bifunctor, in particular the associated Floer complexes are isomorphic after a figure eight correction of the bounding cochain. An appendix with Felix Schm\"{a}schke provides examples of nontrivial figure eight bubbles.