The open problem of calculating the limiting spectrum (or its Shannon transform) of increasingly large random Hermitian finite-band matrices is described. In general, these matrices include a finite number of non-zero diagonals around their main diagonal regardless of their size. Two different communication setups which may be modeled using such matrices are presented: a simple cellular uplink channel, and a time varying inter-symbol interference channel. Selected recent informationtheoretic works dealing directly with such channels are reviewed. Finally, several characteristics of the still unknown limiting spectrum of such matrices are listed, and some reflections are touched upon.