The induced-subgraph counting hardness conjecture

Let $\Phi$ be a computable graph property. For a graph $G$ and positive integer $k$, let $\#{\textsc{IndSub}}(\Phi)$ denote the problem of counting the $k$-vertex subsets of $G$ that induce a graph satisfying $\Phi$, parameterized by $k$. Suppose that there are infinitely many positive integers $k$ such that $\Phi$ is neither true nor false on all graphs with $k$ vertices.

Induced-subgraph counting hardness conjecture. Then $\#{\textsc{IndSub}}(\Phi)$ is $\#{\mathsf{W[1]}}$-hard.

This conjecture proposes an explicit criterion for the $\#{\mathsf{W[1]}}$-hardness of induced-subgraph counting problems. The known dichotomy gives that every such problem is either fixed-parameter tractable or $\#{\mathsf{W[1]}}$-hard, while the conjecture aims to characterize the hard cases; its status is not resolved in the supplied source.