Faber-Harris Conjecture on the Isolation Lemma

For an inclusion-free hypergraph on $n$ vertices, a weight assignment $w:[n]\to[d]$ is isolating when a unique edge attains minimum weight. Faber and Harris conjectured that the number of isolating assignments is at least $n\sum_{j=0}^{d-1} j^{n-1}$, attained by the hypergraph of $n$ singleton edges. The bound holds, and extends to a more general class of objective functions.