"You have things ready?"
"Just as you wanted, father"
As soon as the duke heard that the prince had some dirt on him, he was ready. He has a case against the high court to make the 13th prince lose a significant place in the race for the throne.
The duke's aim wasn't high because he could never aim that high, getting a prince imprisoned is impossible. The evidence is too good enough to put him behind bars, but it is good enough to make him lose face.
The case against the prince surrounds the fact that he was supporting a house of heretics. With a bit of digging, he found that Onia was from a declining household and quickly found that Theia was a heretic.
Supporting a heretic is a serious crime with strict consequences. Most people end up in jail or being executed. They will have a case when they argue that Onia has no right to be in the mobile prison instead of Theia, the true heretic.