Origin of the boson peak in glasses
In plain words
Glasses have more low-frequency vibrations than a crystal would, which shows up as a bump in the vibration spectrum near one terahertz called the boson peak. Which structural feature creates this excess is disputed.
Precise statement
In amorphous solids the ratio $g(\omega)/\omega^{2}$ (vibrational density of states over the Debye law) peaks at $\omega_{\mathrm{BP}}/2\pi \sim 0.5-2\,\mathrm{THz}$ (about $2-8\,\mathrm{meV}$), near the transverse Ioffe-Regel frequency, and correlates with the plateau of thermal conductivity near $10\,\mathrm{K}$. Decide between heterogeneous elasticity (random elastic constants), quasi-localized soft modes with $g \sim \omega^{4}$, a smeared van Hove singularity of transverse acoustic branches, and marginal-stability explanations, by a quantitative prediction of $\omega_{\mathrm{BP}}$ and the peak height across glasses.
What would settle it
A theory predicting $\omega_{BP}$ and peak height from measured elastic constants and structure for several glasses, verified by inelastic neutron or x-ray scattering under pressure or densification.