What is the real relevance of SIC-POVMs (symmetric informationally complete POVMs) to concrete tasks in quantum information theory? A lot of work has been put into giving explicit constructions, and yet it seems that in many concrete applications that at first glance require SIC-POVM-like objects, one can do without them using e.g. suitably chosen set of random projectors. So, my question is - if the problem of constructing SIC-POVMs in arbitrary dimensions was solved one day, would that enable solving problems which can't be solved or approximated using different techniques now?
SIC-POVMs seem to play a role in some approaches to quantum foundations (see e.g. Chris Fuchs, Bayesian approaches to QM that rely on the existence of SIC-POVMs), but it's unclear to me how "mainstream" such applications are.