She [topic-は] argument [possessive-の] during sleeping pretense [object-を] did.
~ふりをする means 'to pretend to ~' — ふり (振り) means 'pretense' and takes a verb in the ~ている form before it: 眠っているふり = 'pretense of sleeping.' 論争 (ろんそう, 'argument, dispute') is a formal word. 間 (あいだ) marks the duration.