probably just systematic error (these things nearly always are), but it's still an exciting result. if another independent experiment confirms, that'll be a huge step.
Do you have a sense of what they need to get from a 4 sigma confidence to 5 sigma?
totally take this with a grain of salt since i'm an astro dude and not a particle physics dude, but usually it entails either more data, or more independent experiments. the former is only feasible if more data will substantially increase your signal-to-noise ratio, and i'm
guessing they've basically maxed that out with the fermilab experiment (i.e., maybe they could run it for another billion years to get to 5σ, but ain't nobody got time for that). the latter is almost certainly the way to go.
basically you can either try to shrink the error bars (more data), or you can add more data points (more experiments). shrinking the error bars is probably not feasible, so they'll need to add more data points with new experiments. lol of course sometimes that only deepens the mystery, like the current tension between independent measurements of the hubble constant.
more than likely, one of these sets of measurements is being affected by an as-yet-unknown systematic error. for example, if the planck data is systematically underestimating H_0, then these measurements may in fact completely agree. so sigma doesn't tell the full story here since sigma is "unaware" of systematics. the same could be true for the brookhaven/fermilab results. unknown systematics may be causing them to overestimate a_μ, and no new physics are needed to put these results back in line with the theoretical predictions.
systematic error is a real asshole.