I am not at all sure whether “active information” would be such a test.
Using “WEASEL” programs as an example, changing the genetic algorithm so that it randomly mutates characters to one of AEHIKLMNRSTW_ instead of to any of A…Z_ would add approximately 1 bit of “active information”, but subsequent mutations would still be “random with respect to function”.[1] Similarly, limiting mutations (either completely or largely) to a subset of the genome or changing the frequency of types of mutation could also add “active information”, but the mutations could still be “random with respect to function".
[1] If you would argue that they are no longer “random with respect to function”, then mutations in the original algorithm aren’t “random with respect to function” either, since they can’t produce all of the characters that could be present in a Shakespearean quote (?:]'.![-";ï,) let alone any of the characters that Shakespeare might conceivably have used (but didn’t) or characters that he wouldn’t have used but could be included as possible mutations anyway.