Yes, your customers will need to enter payment details again for the upsells added via the post-purchase offer. Please see the video below showing the flow. You can also test it yourself on our preview store or by creating a test order.
We plan to update this behavior and allow true one-click upsells once Shopify allows all developers to use the so-called Argo Checkout API.
Happy upselling,
Your Digismoothie Team