Edit January 19: FLI has made a substantive statement about this issue:
https://futureoflife.org/rejection_statement/
This seems concerning. It is claimed that the Future of Life Institute, run by MIT professor Max Tegmark, offered but did not pay out a grant to a Swedish far-right foundation. The character of this foundation and its associates is well-known in Sweden. Expo is an old and respected watchdog organization specialized on neo-nazism and related movements.
https://expo.se/2023/01/elon-musk-funded-nonprofit-run-mit-professor-offered-finance-swedish-pro-nazi-group
I think it very likely that FLI would have made a statement here if there were an innocent or merely negligent explanation (e.g., the document is a forgery, or they got duped somehow into believing the grantee was related to FLI's stated charitable purposes and not pro-Nazi). So, unless there is a satisfactory explanation forthcoming, the stonewalling strongly points to a more sinister one.
On the other hand, U.S. state charity regulators and the Internal Revenue Service's exempt-organizations staff are badly underfunded. Supporting neo-Nazis is constitutionally protected in the U.S. and poses no discernable legal risk. Using charitable resources to promote ends unrelated to the charity is, however, not protected. By making a case much easier to prove with minimal investigation, a written confession would significantly increase the odds of action against the organization and/or personal action against those responsible. In particular, the allegations here if true would support an inference that FLI used charitable resources to further the interests of a family member of the president and treasurer.
As far as PR strategy, do you think FLI could even survive a confession in anything close to its current form? I think the first rule of running a charity is not to embarass your megadonors by bringing them into public scorn by associating them in any way with neo-Nazis. FLI could have felt that this could blow over and remain mostly limited to a foreign publication, while an admission to supporting this organization would be a death knell for FLI.
I'm not attempting to justify their PR decisions as correct, of course. There are higher moral priorities than protecting one's organization. But the PR decisions don't seem irrational on their face if the main/only priority is protecting FLI.