Given that the math community (incl. those 25 Fields medalists) has come out strongly against this trophy hunting of their unsolved problems, to the detriment of mathematics, I don't think these companies are going to be getting positive press if they ignore this plea and continue with this, nor is this going to help turn public sentiment pro-AI.
Presumably the people that OpenAI and Anthropic are trying to impress with these trophy kills are potential IPO investors, but I would have thought investors would also be concerned about the growing public backlash against AI.
They aren't going to sit on millenium solutions regardless (so if Hodge and /or BSD is really done it will get announced especially because of the baseless accusations), and they aren't going to stop trying to solve P/NP and Riemann. The letter doesn't really matter. It's not the first time, and I don't think AI's dramatic ramp in capabilities ever had positive reception from the bulk of mathematicians anyway.