I'm trying to get CERN to use the LHC for a modern particle physics equivalent of the Michelson-Morley experiment. Sent it to the Council Secretariat earlier this week, haven't heard anything yet so may need the publication route.
And I've been brewing on some ideas concerning black holes for a few years.
Still, it would be great to do something like this at the LHC and some of the reasons are explained below.

I'm going to write this proposal in a more scholarly form for publication.
As it should be of great interest for the philosophy of physics. Not to mention the relevance for philosophy of science and mathematics.
That an agenda with promise of a zero result is a good test
for the integrity and methods -- any unknowns that may be hidden by
bias given the nature of building a system that primarily studies noise. Systematic errors is one thing --- systematic human errors another.
And so is the complications from adhering to working theories with
so much unknown.

There is a myriad of theories that deal with space as a variable.
Any form of direction or a experimental zero-proof would settle
many arguments revolving fundamental premises of particle physics.

By actively saying one should do PbPb for one year, and specifically:
That our orientation around the sun related to milky way is the focus of study to look for any space-time effect on any particle. This is not a narrow search in any one area of interest to physics. PbPb as the best option for the amount of tracks and chances for increasing accuracy with more particles of the same type per event. It should be of interest to look at other physics opportunities that would
fit within these run parameters.

It would requires careful consideration of velocity and distance in an angular trajectory for all types of particles over a long period of time.
The influence could be very faint and the more accurately measured to zero, the better an argument theoreticians have for venues in mathematical physics.
In addition to being a good calibration test for the experiments involved.
Of utmost interest is if force carriers may be influenced, and if it will be an expected zero result or if folds or pockets in space discussed for the very small also applies to the very large as is expected.

I've worked at CERN full time in the past, and part time until Desc. 2016.
Sort of dropped out of cognitive sciences with a desire for a philosophy specialisation in 2010.