Thronebreaker: false plumage sidequest

Anyone else having severe difficulties with this quest? I have cleared the entire map except for this - after seemingly winning the opponent continuously draws cards until they have more than I do - Every. Single. Time.

I dont see any cards with affects that relate to this at all - anyone experiencing this or have any suggestions? It feels super broken but I must be missing something.

Thanks - on xbox btw if that matters