When viewing the Object Page>Connected tickets tab - one can select to show Unresolved Tickets, select a saved JQL filter and search string within Summary or Description.

Insight has an option to customize the Object View:
Object Type>Configure>Configure Page Layout

When customizing the Page layout, one can add sidepanels / tabs, and assign them with widgets.

The Connected Tickets widget has no configurable option, except for "Number of Items per page", and loading option.

As a customer - I would like to be able to set a JQL filter on this Tab/Panel - so whenever we access objects in this Object Type, the Tab displays all connected tickets / issues already filtered by the pre-selected JQL filter.