Thanks Neil. Although my interest is more on algebraic triangulated categories, your information should be helpful. Hopefully, some of the proofs on the two references that you mention admit some adaptation.

Thank you Lennart. Your comment is very helpful for me. I had the impression that going to the context of infinity categories or that of (Grothendieck) derivators, the answer might be 'yes'. But still, without adding further structure, I would like to find examples beyond the derived categories of rings (with several objects), for which the answer to my question is positive.

Dear Jeremy, Thank you very mucy for adding TeX formatting to my question.