Here's how our current Stripe implementation works -- we synchronize your "Balance Transactions" from Stripe that effect your payout balance -- charges, refunds, payments, etc. We then take chose charges, refunds, payments, and find the associated entities such as a Stripe invoice or Subscription, and we then enhance the charge, refund, payment with the line items, tax rates, etc from the subscription and/or invoice on Stripe. So we absolutely synchronize invoices, but only when a cash event occurs such as a payment. This means your unpaid invoices in Stripe won't be synced to Greenback until a cash event occurs. This is essentially, cash-basis accounting of your Stripe data.
However, support for unpaid invoices is coming!