My rationale is this:
1. Volantis is a slave state with extremely high levels of slaveholding, and an extreme paranoia about slave revolts.
2. the Old Blood restrict office-holding to bar the children of freedmen.
3. the Old Blood restrict parts of the city to bar the children of freedmen.
4. According to the World of Ice and Fire, the freedmen (and presumably their children) live on the western bank, where according to WOIAF “”precious few voters west of the river.”
Thus, it seems likely to me that the city’s ban continues.
To give a historical analogy, the children of freedmen weren’t allowed to vote in the antebellum South.