I'd think the answer to that would have to be yes.
When a council is evenly split, the chairman has a duty to settle the matter through the use of the casting vote. There is a convention that the chairman uses the casting vote in favour of the status quo, but it is only a convention.
If the person involved is chairing the meeting as retiring chairman from the previous council, then it could be argued that a vote for themselves was indeed support for the status quo.
Other cases could be argued on their merits, but in the end, the chairman is entitled to use the casting vote as they see fit. Either way, the unfortunate fact is that half the council is going to be dissatisfied.