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We present measurements on an aperiodic device-specific longitudinal-mode pattern in InGaN laser diodes. The characteristic shape of this pattern occurs only if the laser is driven slightly above threshold; in addition, it tunes with temperature at exactly the same rate as the cavity modes. By careful selection of the collection optics and averaging ten rapid scans over 15 min in a high-resolution Fourier transform spectrometer, we could exclude possible explanations like beating of mode families, self-pulsation, or external reflections. A naive simulation of the longitudinal modes profiting from their individual "gain profile" along the cavity suggests that we see the signature of quantum-well thickness fluctuations.