  For a compact set $K$ with contact type boundary in a symplectic manifold $M$
we construct a spectral sequence from the local Floer homology of the Reeb
orbits, as studied by \cite{Mclean2012}, to the relative symplectic cohomology
of $K$ in $M$ over the Novikov ring. The spectral sequence is functorial with
respect to inclusions which are not required to be exact. This functoriality is
key to the closed string reconstruction problem near the singularity of an SYZ
fibration. We illustrate this in the case of dimension $2n=4$ for symplectic
cluster manifolds. In higher dimension, an additional ingredient, the locality
spectral sequence, is required, and is the subject of a forthcoming work in
progress.
