Products in a users cart that goes out of stock

Within our e-commerce solution we have a stock indication for in, out and low on stock. It's B2B and the use case exists that a user might start filling their cart, leave the website and come back later.

The problem here is that some products that the user has in the cart can go out of stock. I am fiddling with how to handle this because removing the product does not seem like a nice thing to do. The discussion now happens whether we should:

  1. Still display the product with an out of stock message and let the user continue.
  2. Grey out the continue button in the shopping cart before the user removes the out of stock product from his cart.

Does anyone had to work with this case before?