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Generally speaking, yes. That would in the vast majority of cases be an ethical conflict of interest, and may serve to void the tax exempt status of the not-for-profit entity. The only way that this would not be considered a conflict of interest is if the elected member refrained from voting on the hire and was not present at the time of the vote. That said, even then, this would likely be, and probably would be considered a conflict of interest and should not occur.
A formal protest can and should be launched by an interested party with the Board of Trustees relative to the issue with the stated purpose being to void the hire. The mechanism to challenge should be in the operating agreement or procedures of the Board.
Let me know if you have any other questions or comments.
Best wishes going forward.