In the standard cosmological model, where space expands according to the FLRW metric under the influence of $\Lambda$CDM (dark energy, $\Lambda$, and cold dark matter).
In the standard model of particle physics the masses of the fundamental particles are set by the vacuum expectation value (VEV) of the Higgs field and the strength of the particle's interaction with that field.
So, the question is this: how do we disentangle the effect of an expanding universe from the effect of a Higgs field VEV that is changing with time? Specifically, an increasing Higgs VEV shrinks the physical size of atoms, increasing the frequency of their transitions, making existing photons lower energy, by comparison.
I'm especially interested in observable differences that have been observed or looked for because both a changeable Higgs VEV and dynamic space-time are predicted by separate, well supported, theories (the standard model of particle physics and general relativity, respectively). For instance, is there a difference in how the cosmic microwave background would appear? Or would the VEV changing alter the proton's mass in a way that keeps $m_e / m_p$ constant (affecting hyperfine transitions)? The latter is not a trivial question because most of the proton's mass comes from gluons at a scale set by color confinement and that, as far as I know, has not yet been connected to the Higgs VEV.