Does every sparse random graph model satisfying a zero-one law also satisfy a convergence law for all sentences of first-order logic with counting quantifiers?

The classical zero-one law for dense Erdos-Renyi random graphs tells us that every first-order sentence is almost surely true or almost surely false. For sparse random graphs, the situation is more delicate: some models satisfy zero-one laws only for certain fragments of logic, and others satisfy only convergence laws where probabilities converge but not necessarily to 0 or 1. The open problem is whether sparse random graph models that are known to satisfy zero-one laws for basic first-order logic automatically extend this behavior to richer logical languages that include counting quantifiers, which allow statements like 'there exist at least k elements satisfying some property.'