An interesting, if slightly misinformed link there Mibs with regards to the clock synchronisation. I must admit I didn't read the bit about the neutrino path being through rocks and assumed it would be possible to use another medium to test for clock offsets.
The link you give speaks about the effect of relativity on the GPS clocks but timing receivers use multiple satellites (and multiple constellations such as GLONASS) and long-term averaging of the received timecodes to reduce the effects that this, and other clock errors on individual satellites, has. If the ground clocks are moving away from some satellites then they are moving closer to others, making it easy for the receiver, which has a stable internal reference itself to see that this is happening.
It is also possible to synchronise cesium clocks with each other and, using battery power supplies, to transport them to the individual sites, this will elminate the reliance on GNSS systems to provide the timing reference. I would assume this has already been done, if not then that would be my next move.
However, my original point was that 60ns is a large offset and the mechanisms to synchronise clocks are well understood so I would be surprised if it turns out that clocking errors are source of the anomalies here - as opposed to measurement errors which I do not know anything about!