I'm a graduate student studying QFT. I'm quite interested that is color confinement detected or proved? (both directly and indirectly) Or it is just an assumption?
Here is an experimentalist's answer.
Color confinement is a theoretical concept arising from the plethora of experimental observations that are summed up theoretically in the Standard Model. We have no free quarks or gluons, we do have quark jets and gluon jets. So confinement as predicted by the SU(3)xSU(2)xU(1) SM is consistent with all the existing data.
One has to keep in mind that a theory applying to experimental data can be falsified,or can be found consistent with the data; but consistent is not proof, it is a temporary validation.
A theory of course has axioms and mathematical proofs, so a theorist should answer whether the theory of QCD allows unconfined manifestations of color. These should be in phase spaces not explored by present experiments.