Local-global convergence versus large deviations convergence for bounded-degree graphs

Determine whether local-global convergence of sequences of finite graphs of uniformly bounded degree implies large deviations convergence of such sequences. Prove the implication or produce a counterexample.

Background

The paper draws an analogy between strong-quotient convergence of representation sequences and local-global convergence of graphs. It further relates its entropy convergence to large deviations convergence. In combinatorics, the precise relationship between these two convergence notions remains unresolved.