Now I’ve made some new plots using Tim’s exact proposition 4 in the wish list: choose either -1 or +1 at prime p depending on which minimizes the discrepancy up to the next prime q.
First here is a plot (click to enlarge) starting with initial condition only.
Here is another plot using as a seed the first 1124 sequence.
Here is the same seed over an even longer interval.
For comments and discussion please see the wiki and links there.