According to Aristocra Formbi, the Third Vagaari War caused shortages in Chiss manpower, and to counterbalance this, the Chiss contracted Killik hives to work for them. Although a series of safeguards were put in place to prevent the Killiks telepathic abilities turning the Chiss into Joiners, these were compromised, leading to the absorption of two of the nine Chiss Ruling Families into the Killik nests.
At this point, a dispute arose between the surviving Families, three of which had become heavily dependent on the Killiks: these Families presumably wished to continue making use of the Colony or else to receive some form of compensation, but the other four Families resisted, and entered into an alliance with the Household Phalanx. Thus reinforced, the Four Families destroyed their rivals, leading to the new political situation encountered by the Jedi expedition which visited the Chiss Ascendancy in 28 ABY.
However, new evidence now suggests that no conflict between the Chiss and the Vagaari had taken place before Luke Skywalker and Mara Jade Skywalker visited the remains of the Outbound Flight Project in 22 ABY: the subsequent Battle of the Redoubt and the Chiss attack against the Vagaari fleet represented the first open hostilities between the two peoples, and it is unclear whether three wars with the Vagaari, plus the extended aftermath which Formbi attributes to the Third Vagaari war, could have occurred between then and 29 ABY.
It is thus uncertain whether the Third Vagaari War ever really happened, or how reliable any of the information supplied by Formbi was.