WI War between Eastern and Western Roman Empire in 421 AD?

In 8 February 421 Emperor of the West Honorius raised General Constantius to co-Emperor along (he was married to his sister Placidia). However this appointment wasnt well received by the Emperor of the East Theodosius II (Honorius's uncle) and he refused to recognise him.
Constantius III prepared to take military action against Theodosius II in order to force him to recognise him as a lawful Western Emperor, but he died suddenly in 2 September of the same year (well not without suspicion of foul play).
WI Constantius III didnt die and attacked Theodosius II? What would be the consequences of a war between the 2 empires?
 
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