Axanar Productions has been continuing work, business as usual, getting things ready for shooting.
While 'no comment' is the usual mantra once you enter into the lawsuit arena (i.e. don't give the opposition any more ammunition) I haven't seen much in the last couple of weeks that would indicate that an amicable settlement is in the cards. I do think that this is the only way that Axanar can move forward.
As mentioned above by a couple of you, this looks like 'image control' to me. Fans (and at least one TOS alum, George Takei) have already been expressing skepticism r.e. the upcoming movie, and the studio execs may not want to have Axanar out there as something to make their own production look bad. Unfortunately, CBS/Paramount have went the Lowest Common Denominator route with the last two movies, going for 'flash' over 'substance'. Axanar looks to be a more intellectual exercise (based on the tone set in Prelude), focusing more on the story and less on the 'pew pew'.
So in short, IMHO CBS/Paramount don't want Axanar making their own efforts look bad... otherwise they'd go the 'amicable' route and work out licensing. Of course, we don't really know, and unfortunately the legal system route can often take years to work out a resolution.
The change.org petition (there are actually 3, but only one is getting traction) seems to be doing OK. 11,216 supporters so far as of this post:
https://www.change.org/p/cbs-support-axanarThe other two petitions (as of this post, these have less than 100 sigs each)
https://www.change.org/p/cbs-save-star-trek-axanarhttps://www.change.org/p/columbia-broadcast-network-and-paramount-pictures-tell-cbs-and-paramount-that-they-must-drop-their-lawsuit-against-axanar