Setting default widget options

I would like to set additional options for a widget upon registration and have those options propagate to existing widgets of that type.  I have this code in my plugin_init(), where the definition includes 'option_name' => 'option value'.


         'id' => 'my_widget'

        ,'name' => 'my widget'

        ,'description' => 'My widget'

        ,'context' => ['dashboard']

        ,'option_name' => 'option value'


I thought I would find the new option_name and 'option value' in the private_settings table for the associated widgets, but I don't.  Neither do I see it as $widget->option_name value.  I thought (hoped) that propagation to the widget would happen when Widget Manager's applyWidgetsConfig hook handler triggers.

Do you have any advice for setting and getting default widget options?