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

RSFT functors for strong cobordisms and applications