Yes, your boat must remain insured and hold a valid BSC. The boat still belongs to you, meaning you are still responsible for it so it would be within your best interest to have the boat fully insured.
Having a valid BSC makes the boat more appealing to prospective buyers, as they know that it has been inspected and meets the relevant safety guidelines. It is also peace of mind for us knowing we are selling a boat that is safe and up to standard.