This can be done using the "window action". You can set the value of "context" under the action and achieve this.

Try to find the "action" for that menu and try to define your filter under the "context" attribute.

See the example

Product under Sales menu,

context = {"search_default_filter_to_sell":1} ---> this leads to default selection of "sellable" product filter.

context={"search_default_filter_to_purchase":1} ---> this leads to default selection of "purchase" product filter.

So here in the search view there is a filter already defined as "filter_to_sell" and "filter_to_purchase" and using this in the context value for different actions, appropriate filters are set as default selection during load of the list view.

For your reference in the product module search for "product_normal_action_sell" this action, you will get the idea.

So same way you need to define your filter and use the context attribute of the action.

Hope this helps !!.