Will the bundles keep inventory sync?

Yes, for both variant and discount code modes.

The variant mode

For the variant mode, when an order is paid, the individual items are added as free items to the order. The inventory of the individual items will be deducted. So the customers will see the individual SKUs appearing on the order within ~10s. 

However, the following situations may lead to an overselling issue:

1. if the customer adds a bundle to the cart but doesn't check out for a few days and one of the items selected is sold out
2. Many customers create bundles and checkout at the same time. But the stocks of the items are very low

BYOB relies on Order Edit to keep inventory in sync. However, there are some limitations to prevent order edits from proceeding. Please refer to this article for more info: Why aren't my orders edited?

The discount code mode

Only the individual items are added to the cart. Everything is Shopify-native. So there is no problem to keep inventory in sync