remove null values from json

This is ALMOST what I want:
select json_strip_nulls ('[{ "f1":1 , "f2":null , "f3":"NULL" , "f4":""
}]');
the result is:
"[{"f1":1,"f3":"NULL","f4":""}]"

As you can see, the "f2" object has been removed. I would *also* like to
remove the f3 and f4 objects.

I realize I won't be able to use the json_strip_nulls function to do this.
I could brute force it by doing "json to array" then looping through the
array and picking only the objects I want, and then using the chosen
objects to build an new JSON document.

I suspect there is a more elegant way to accomplish the same objective.
Ideas?

PS. It would be great to have a json_strip_values (json, array) where array
contains the list of values you want removed.