Products not being removed nor unpublished from catalog after feed synchronization

Hi,

I’m currently setting up a custom catalog feed on Klaviyo. Because of the number of products we have, they’re splitted into multiple catalog sources.

My question is: how is a product not sold anymore supposed to disappear from the catalog? I’ve tried to sync with almost empty catalog sources to see if almost all my products would disappear or be unpublished, but I still see all the products in the catalog, with the status “Published”. Is there a delay or something before the products from the catalog actually reflects what’s in the synced sources?

Thank you in advance for your help.