Be filter removed. Note that the intensity of (110) has been magnified by a factor of 10 relative to the (004) data.](lattice){width="1\columnwidth"}

 \[sec:energies\] Phonon dispersion
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We first describe the softening in energy of the acoustic fluctuations in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}. Inelastic neutron scattering experiments were performed using the triple-axis spectrometer (TAS) EIGER [@Eiger2017] at the spallation source SINQ [@Fischer97] at the Paul Scherrer Institute (PSI) in Villigen, Switzerland and the TAS IN22 [@Regnault1999] at the Institute Laue-Langevin (ILL) in Grenoble, France. Both spectrometers were operated in the constant final energy mode with $E_f = 14.68\,\mathrm{meV}$. At EIGER and IN22, the dispersion of the acoustic phonon branches was measured using a vertically focusing monochromator and horizontally focusing analyser without using any collimation. For the EIGER measurements, crystals oriented in the $hk0$ and the $hhl$ planes were used, collecting data around the $(400)$ and $(440)$ Bragg peaks. For IN22 we used the $hk0$ scattering plane.

In addition, experiments at small reduced momentum transfer $q$ were conducted using the cold TAS option of MIRA [@MIRA; @MIRAnew] at the MLZ in Garching, Germany, using neutrons with fixed incident energies $3.78\,\mathrm{meV} < E_i < 4.97\,\mathrm{meV}$ yielding an energy resolution (full-width at half-maximum, FWHM) of $0.11\,\mathrm{meV} < 