The problem with all forms of "random" structures - whether Brownian, Chaitin's Omega, etc. - is that they are at root the effect of a prior cause and thus, from a perspective outside of the hypercube of space/time, or from the perspective of an extra time dimension within the hypercube, knowable and thus, non-random or as Wolfram says, pseudorandom.
This has not been demonstrated. In QM, such a demonstration would contradict some experiments. Were such a cause known (or in QM, known to exist), there would be measurable consequences. How would one go about showing the existence of a "cause" that mimicked "randomness"?
(I'm not sure that Chaitin's Omega is "randomly generated" in common usage. It is very complex, as are randomly generated strings.)