Uniqueness of difficult subgraph-counting identities

Let a subgraph-counting identity be an identity of the form described above, involving induced-subgraph counts in the neighborhoods and non-neighborhoods of vertices, and classify identities by quotienting out the easily described families and the difficult identities for lower values of $K$.

Uniqueness conjecture. For every $K\geq4$, there is only one subgraph-counting identity modulo the easily described families and the difficult identities for lower $K$.

This conjecture asserts uniqueness at each complexity level after the stated equivalences are imposed. The paper does not provide a proof or resolution of this claim.