Thank you for posting this question! To answer it, yes, you can continue to display your Stories widget on your website even after your stories expire on Instagram.

To do this, please follow these steps:

1. Login to your account, open the desired widget, and on the settings on the right click 'General settings':

2. Scroll down and click 'Yes' on 'Show widget after story expires':

That's it! Your stories will continue to show on your website even if they expire on Instagram.