You can go the CSS route. Create a new CSS class (in this case dropdown-no-empty), then define it to hide the first option tag under select tags under objects with the new class.
Note: This is in SCSS.
Assign the class to your reference selector.
You could use the format JS widget to remove it.
Refer to the dropdown selector and remove the empty value.
See this idea for some code you can use in an HTMLSnippet widget: