The File Geodatabase API documentation
README states (near line 159):
While the Esri File Geodatabase API supports reading the schema and
data of complex geodatabase types, the API does not honor geodatabase
behavior on inserts, deletes or updates to the following dataset
* Annotation and Dimension feature classes
* Relationship Classes
* Networks (GN and ND)
* Parcel Fabrics
This seems to indicate that you can read the base geometry from the Esri API, but there isn't any access to the ArcObjects network topology components (connectivity) through FileGDBAPI DLL.
Since the Esri API does not support it, the reverse-engineered API clone isn't likely to, either, making it far more difficult for QGIS to provide it (with either implementation).
Esri's purpose in releasing the API was to support read/write of simple data in file geodatabase format, to facilitate open source tools' manipulation of file geodatabase data. Implementing more complex behaviors would have greatly expanded the scope of the effort, and greatly increased the code which would have had to be rewritten to support the release of the library.