Duke Balzac wasn't foolish, neither was he uninformed of the rumors floating about in the capital. Even though he might appear indifferent on the outside, as the duke, the man had his own source of information. But, he chose to stay quiet about it because the rumor about his niece, Ophelia, had played to his own advantage.
By putting the young lady in the spotlight, the duke thought he could prevent any real danger from befalling his youngest daughter. Yes, many would jeer at Alice but nobody would take her seriously either because of the rumor that said the duke favored Lady Ophelia more than Lady Alice.
The duke thought since the opportunity had presented itself than he should use Ophelia as his daughter's shield well. Thus, he chose not to comment about it and let the rumor lingered as long as Ophelia didn't do anything outrageous. He needed to keep her for the spotlight but most importantly for the church's eyes.