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She'll probably carry out her plan regardless of the expense.
Literal
She [topic-は] expense [regardless of-に関わらず] own plan [object-を] will-carry-out [probably-だろう].
~に関わらず (regardless of) dismisses the expense as irrelevant to her decision. 自分の計画 (her own plan) uses 自分 to emphasize that it's her personal plan. 実行する (to carry out, to execute) is a formal compound verb for putting plans into action. The sentence-final だろう adds conjecture — the speaker believes this will happen but can't be certain.