I had a very similar case: 4 out of 9 randomized question groups should be displayed (you should need just to change 4 to 1). After a long search I didn't find any useful solution, so I created one with one ugly side effect:
put all question groups in the same randomization group
choose one question in each question group, e.g., QG1Q1, QG2Q1 etc.
use these questions in the corresponding question group for the "group relevance equation" like
in question group 1: "QG1Q1.gseq LT 4"
in question group 2: "QG2Q1.gseq LT 4"
…
Outcome: your question groups get displayed randomly. The gseq functions for a question returns the question group sequence number. That is, if your survey starts with your randomized question groups, gseq starts with 1. After 4 question groups the next question group in your randomization group will not be displayed and skipped. In your case use 1 in the equation instead of 4.
Ugly side effect: the progress bar is not working anymore!
@tpartner: should I file a bug or do you have any other option? Our LS version is 2.05+ Build 141210.