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Ever since they were first detected over 100 yr ago, the mysterious diffuse interstellar bands (DIBs), a set of several hundred broad absorption features seen against distant stars in the optical and near-infrared wavelength range, largely remain unidentified. The close match, both in wavelengths and in relative strengths, recently found between the experimental absorption spectra of gas-phase buckminsterfullerene ions (C$_{60}^{+}$) and four DIBs at $\lambda 9632\, {\rm \mathring{\rm A}}$, $\lambda 9577\, {\rm \mathring{\rm A}}$, $\lambda 9428\, {\rm \mathring{\rm A}}$ and $\lambda 9365\, {\rm \mathring{\rm A}}$ (and, to a lesser degree, a weaker DIB at $\lambda 9348\, {\rm \mathring{\rm A}}$) suggests that C$_{60}^{+}$ is a promising carrier for these DIBs. However, arguments against the C$_{60}^{+}$ identification remain and are mostly concerned with the large variation in the intensity ratios of the $\lambda 9632\, {\rm \mathring{\rm A}}$ and $\lambda 9577\, {\rm \mathring{\rm A}}$ DIBs. In this work, we search for these DIBs in the X-shooter archival data of the European Southern Observatory’s Very Large Telescope, and we identify the $\lambda 9632\, {\rm \mathring{\rm A}}$, $\lambda 9577\, {\rm \mathring{\rm A}}$, $\lambda 9428\, {\rm \mathring{\rm A}}$ and $\lambda 9365\, {\rm \mathring{\rm A}}$ DIBs in a sample of 25 stars. While the $\lambda 9428\, {\rm \mathring{\rm A}}$ and $\lambda 9365\, {\rm \mathring{\rm A}}$ DIBs are too noisy to allow any reliable analysis, the $\lambda 9632\, {\rm \mathring{\rm A}}$ and $\lambda 9577\, {\rm \mathring{\rm A}}$ DIBs are unambiguously detected and, after correcting for telluric water vapour absorption, their correlation can be used to probe their origin. To this end, we select a subsample of nine hot, O- or B0-type stars of which the stellar Mg ii contamination to the $\lambda 9632\, {\rm \mathring{\rm A}}$ DIB is negligibly small. We find that their equivalent widths, after being normalized by reddening to eliminate their common correlation with the density of interstellar clouds, exhibit a tight, positive correlation, supporting C$_{60}^{+}$ as the carrier of the $\lambda 9632\, {\rm \mathring{\rm A}}$ and $\lambda 9577\, {\rm \mathring{\rm A}}$ DIBs.