If he doesn't resign in the next week- it will be too late to call a special election for his seat on general election day- forcing the special election to be called on its own. There are multiple problems with this.
First, a special election in December will draw significantly less people to the polls compared to the general election in November. No one benefits from limiting participation.
Second, there are costs involved in a special election. Besides the money to program voting machines, pay poll workers and all the normal election costs, there are also the costs of taking up rooms in the many polling places around Prince William County. Since many of the precincts are at schools- if the election is held in the gym, the students won't get P.E. that day, if the election is held in the music room, students won't have music class- etc.
Third, special interests can have a much bigger impact in special elections by spending money to turnout voters. In a general election their money has to be spent on persuasion, which has questionable impact from interest groups, while a special election plays to their strength of mobilizing voters.
Personally, I think some of the attacks on Sean Connaughton's fiscal responsibility in Prince William have been misplaced, but if he doesn't resign this week and allow a special election in November then he will have given his critics another chance to cite unnecessary spending, and that would be a shame.