  We extend the hierarchy functors of [30] for the case of strong symplectic
cobordisms, via deformations with Maurer--Cartan elements. In particular, we
prove that the concave boundary of a strong cobordism has finite algebraic
planar torsion if the convex boundary does, which yields a functorial proof of
finite algebraic planar torsion for contact manifolds admitting strong
cobordisms to overtwisted contact manifolds. We also show the existence of
contact $3$-folds without strong cobordisms to the standard contact $3$-sphere,
that are not cofillable. We also include generalizations of the theory, which
appeared in an earlier version of [30], relating our notion of algebraic planar
torsion to Latschev--Wendl's notion of algebraic torsion, discussing variations
from counting holomorphic curves with general constraints and invariants
extracted from higher genera holomorphic curves from an algebraic perspective.
