Does every sparse random graph model satisfying a zero-one law admit a computable Scott sentence that almost surely describes the limit structure?

The paper on limit laws for sparse random graphs establishes that certain random graph models almost surely satisfy or violate specific logical sentences, producing a probabilistic limit theory. Separately, the paper on computable Scott analysis develops tools for measuring the descriptive complexity of infinite structures through Scott sentences. The open problem is whether these two programs can be unified: when a sparse random graph model obeys a zero-one law, can one always produce a computable Scott sentence whose unique model is, in a precise sense, the almost-sure limit structure, and if so what is the least complexity such a sentence requires?