Distinct topological functionals from inequivalent enriched TQFTs

Let $Y$ be an $n$-finite space, meaning that $\pi_0Y$ and all homotopy groups $\pi_q(Y,y)$ for $0\leq q\leq n$ and $y\in Y$ are finite. An $n$-dimensional bosonic or fermionic $Y$-enriched fully extended TQFT is a theory whose partition function defines the corresponding topological functional to $Y$. Distinct-functional conjecture. Inequivalent $n$-dimensional bosonic, respectively fermionic, $Y$-enriched fully extended TQFTs produce different $n$-dimensional bosonic, respectively fermionic, topological functionals to $Y$. This asserts completeness of partition functions as invariants of these theories, but the source gives no resolution of the conjecture.