Sure the family has the means of ending the auction, but it wouldn't benefit them. Even if they disagree with the process, there's plenty other influential members. If they shut it down, they'll be the target of bad reputation. Nobles have to keep other nobles happy or else their sandcastles...
I'm such a greedy bastard, here they released three chapters in a week (possible thanks to the random citizen??) and here I am asking for one more tomorrow. 😭
All I'm saying is what's wrong with setting a trap someone could avoid by simply not doing the bad thing that will lead them falling into the trap.
If the person doesn't go about their misconduct then they wouldn't have gotten harm. In this case they went ahead and got caught, so what's wrong...