The identity of the carriers of the diffuse interstellar bands (DIBs) is one of the most fascinating puzzles of modern spectroscopy. Over the last few years the number of known DIBs has grown substantially. In this paper we discuss the two recently discovered near-infrared weak interstellar features which have already been proposed as fingerprints of the buckminsterfullerene [formmu2] We present and discuss measurements of the two related DIBs within a larger sample of reddened targets, observed with different spectrometers, telescopes and site conditions. We provide additional arguments in favour of the interstellar origin of the two bands. We find evidence around the 9577-Å DIB of far-wing structures, which may affect broad-band measurements. We estimate corrections and errors for telluric and stellar blends, and show that the cores of the two DIBs are well correlated with a ratio near unity within 20per cent. Finally, we discuss their relation to the laboratory spectra of [formmu3] and the search for two expected weaker [formmu4] transitions.