We study the uniform query reliability problem, which asks, for a fixed Boolean query Q, given an instance I, how many subinstances of I satisfy Q. Equivalently, this is a restricted case of Boolean query evaluation on tuple-independent probabilistic databases where all facts must have probability 1/2. We focus on graph signatures, and on queries closed under homomorphisms. We show that for any such query that is unbounded, i.e., not equivalent to a union of conjunctive queries, the uniform reliability problem is #P-hard. This recaptures the hardness, e.g., of s-t connectedness, which counts how many subgraphs of an input graph have a path between a source and a sink. This new hardness result on uniform reliability strengthens our earlier hardness result on probabilistic query evaluation for unbounded homomorphism-closed queries (ICDT'20). Indeed, our earlier proof crucially used facts with probability 1, so it did not apply to the unweighted case. The new proof presented in this paper avoids this; it uses our recent hardness result on uniform reliability for non-hierarchical conjunctive queries without self-joins (ICDT'21), along with new techniques.