Exponential succinctness gap over a fixed vocabulary

Determine whether an exponential succinctness gap between three-variable first-order logic and the calculus of relations holds for a family of formulas over a fixed signature, such as a single binary relation symbol.

Background

The paper establishes an exponential lower bound for translating positive three-variable first-order formulas with one quantifier into relational circuits over finite structures. However, the hard families Ψ_k and Φ_k use k and 2k relation symbols, respectively, so their formula size is linear in k partly because symbol occurrences are counted as single syntax-tree nodes.

The authors explicitly leave unresolved whether the same exponential gap can be obtained when the vocabulary is fixed in advance, for example when all formulas use only one binary relation symbol. This question concerns extending the established lower bound beyond signatures whose size grows with the parameter.