Exactness of the Chain Complex Construction Procedure

Let the Chain Complex Construction Procedure be the procedure described in the paper for enlarging an initial map by adding all possible diagrams from the set $D$, producing an infinite chain complex of finitely generated free modules.

Exactness conjecture. The chain complex obtained by applying the Chain Complex Construction Procedure results in a free resolution of the cokernel of the initial map. That is, it is an exact chain complex of free modules except at the beginning.

This claim asserts that adding all possible diagrams ensures exactness, rather than merely producing a chain complex. The supplied text does not indicate whether this claim has been proved or remains open.