Product List added as block is not updated by AJAX on filter change

Problem/Motivation

Product List added as block is not updated by AJAX on filter change.

Steps to reproduce

1. Add a block "Product Filters" to the "Sidebar first" with a unique index, eg, 42.
2. Add a block "Product List" to the "Sidebar second" with the same unique index.
3. Open the site frontpage.
4. Choose something in the filter.
The product list block should be updated by AJAX, but it doesn't update.

When fixing, check also another order - the product list on the sidebar first, product filters on the sidebar second.