You don't need to speculate here, since much of the story is not being disputed.
The recent ground-breaking work on Navier-Stokes "blow-ups" was done over a period of years by mathemtaticians Diego C´ordoba and Luis Martınez-Zoroa.
NYU professor Tristan Buckmaster and Anthropic employee (& mathematician) Levent Alpoge took the above work as a starting point, and over a year with LLM assistance developed a blow-up proof under certain conditions.
Buckmaster: "We used several LLMs throughout: Anthropic’s Claude, OpenAI’s Codex, especially with GPT-5.6 Sol and, more recently, Astra. The latter was only used for writeups and auditing our arguments."
Buckmaster says he thinks that Martınez-Zoroa, whose work this all builds on, deserves the Fields Medal for his work.
OpenAI claim that on Sept 1st they heard a rumor the problem has been solved (which happened on August 15th), and then decided to re-solve it themselves using a 2-week old model, then later reached out to Prof. Buckmaster and Levant to come to some agreement to co-publish.
OpenAI: "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models ". In other words, not only did they deliberately choose to tackle a problem they heard had already been solved (in turns out only partially solved), but they may have done so using a model that was aware of the successful way to attack the problem.
It seems there are three potential scandals here:
1) OpenAI by their own admission chose to try to scoop mathematicians who they had heard had already completed a proof
2) OpenAI may have used a model that had seen "de-identified" messages indicating the direction to take
3) An OpenAI employee essentially threatened to "ruin the career" of the NYU professor who had been working on this if he did not cooperate with them
The direct plagiarism possibility, 2), while it should be a warning to anyone using OpenAI's models, doesn't need to be true for OpenAI to have benefited from the researcher's work. It's enough that they heard Navier-Stokes had been solved and could then go out with their swarm of 10,000 agents and $20M of compute to hunt out the latest research and brute force it.
Magnus Carlson once said that if he wanted to cheat all it would take would be for someone to indicate to him (a wink from someone in the audience perhaps) when a position warranted more time to be spent on it (because there was something important to be found if he did). It seems that, at absolute minimum, this is what OpenAI did here, although in context of math this is not cheating - the "wink" was a rumor, originating from who knows where, that a proof existed (but had not yet been published) and therefore there was potential to rush in and scoop rights to publish or co-publish.