Characterization of zero asymptotic variance for component functionals

Let $\psi$ be a graph functional satisfying the paper's growth condition, let $p_T$ denote the relevant probability associated with a finite tree $T$, and let $\sigma^2_\psi$ be the asymptotic variance in Theorem~TGpsi. Assume the degree-distribution conditions -- and the supercriticality and positivity conditions --. Zero-variance conjecture.

\sigma^2_\psi=0\iff \psi(T)=0\text{ for every tree $T$ with $|T|>1$ and $p_T>0$.

The trivial examples in Example~Etrivial show that several contributions can force zero variance; the conjecture asserts that, under the stated assumptions, these are the only such cases. The paper explains that values of $\psi$ on $\mathsf K_1$, trees with $p_T=0$, and non-trees do not affect $\sigma^2_\psi$.