Tellurium solution in Xenon
Liquid xenon is a quite decent solvent (See e.g. https://www.nature.com/articles/293165a0). The chances are that it will be easily able to dissolve the tellurium produced over the lifetime of the apparatus.
If we very generously assume that the experiment will run for 1000 years, then the decay of Xe-124 will produce about 1e6 tellurium atoms over this time, giving the molar concentration of about 1e3 atoms/litre (rounding the liquid xenon density to 3.5 g/cm3), or on the order of 1 zeptomole/litre (1e-21 mole/litre). If we assume that the xenon is kept just above its freezing temperature at the ambient pressure (ie 162K), this concentration of tellurium will remain dissolved as long as the Gibbs free energy of dissolution at this temperature is less than:
-R T ln [Te] = -8.3 J/mol-K * 162 K * ln(1e-21) * 1e-3 kJ/J = 65 kJ/mol
In chemical terms, this is an enormous number - it exceed the heat of vaporization of Tellurium (which is about 53 kJ/mol). I can't be bothered to estimate the free energy of vaporization at 162K, but it will be less than 53 kJ/mol, since vaporization increases the entropy. So, the tellurium will definitely remain in solution after 1000 years.
How long will we have to wait to see it to start precipitating at the bottom? Let's take 53 kJ/mol as the worst-possible-case estimate of the free energy of dissolution. Then, the critical concentration of tellurium will be:
exp[-53 kJ/mol * 1000 J/kJ / ( 8.3 J/mol-K * 162 K )], which is about 10 atto-mole/litre (1e-17 mole/litre)
That's the concentration we'll reach after waiting for 10 million years. Because the actual free energy of tellurium dissolution is xenon is very, very unlikely to be this high (after all, Xenon -is- a good solvent), we'll probably have to wait for much longer. Each 10 kJ/mol decrease in the free energy will increase the waiting time by a factor of 1700 at this temperature - so it won't take much to get us beyond the lifetime of the universe so far.