I haven't looked into LIGO at all, so this is very likely not just off-the-cuff but out to lunch. But I'll ask: What if people start looking at the (very) precise timing of particle beam circulation e.g. at CERN?

I would guess there is not sufficient resolution. And maybe it's beyond what's possible. But, thinking of additional avenues for measurement, and something that might be extant or require less build-out.

And now, there would be something against which to compare anomalies, were they to be measurable.