Existence of difficult subgraph-counting identities

Let $G$ be a graph with $n$ vertices. For a graph $J$, write $s(J,G)$ for the number of induced subgraphs of $G$ isomorphic to $J$; let $G_v^-$ and $G_v^+$ denote the subgraphs induced by the vertices not adjacent and adjacent to $v$, respectively. A subgraph-counting identity has the form

$$\sum_{v\in V(G)} \sum_{a=1}^N m_a(n) \prod_{i=1}^k s(J_{a,i},G^-_v) \prod_{j=1}^l s(J'_{a,j},G^+_v)=0.$$

Here $m_a(n)$ are polynomials in $n$, and the complexity condition is

$$\deg m_a(n)+\sum_{i=1}^k |V(J_{a,i})|+\sum_{j=1}^l |V(J'_{a,j})|\leq K$$

for every $a$.

Existence conjecture. For every $K\geq 4$, there is such an identity that holds for every graph $G$ with $n$ vertices, has one graph $J_{a,i}$ or $J'_{a,j}$ with $K$ vertices, and is not in one of the easily described families.

The paper's proved identity supplies the first example outside the previously known easily described families. The conjecture proposes that difficult identities continue to exist at every complexity level $K\geq4$.