Since you paid this expense, you would be entitled to reimbursement of the cost, before anyone else is able to receive anything. You should be entitled to reimbursement of the entire amount paid, not just the refund.
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The refund will come back to the estate.. the IRS does not know who paid the estate tax origionally..
However, I agree with Attorney Frederick.. your payment on behalf of the state becomes a loan to the estate (e.g. a debt of the estate) you should be entitled to refund of the entire payment (not just the refund) before any distributions to anyone ... and potentially before any other expenses of the estate are paid.
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