Does a zero-one law hold for first-order logic on the giant component of a sparse Erdos-Renyi random graph G(n, c/n) for all constants c greater than 1?

The classical zero-one law for first-order logic on dense random graphs says that every first-order sentence holds with probability tending to either 0 or 1. For sparse random graphs G(n, c/n), the situation is far more subtle. When c is less than 1, the graph consists of small tree-like components, and various limit laws have been studied. But when c exceeds 1, a giant component containing a linear fraction of vertices emerges, and it is genuinely unclear whether first-order logic restricted to this giant component satisfies any clean zero-one law or convergence law, especially after the pruning operations studied in recent work on component-pruned graphs.