How do earthquakes occur at depths of 300 to 700 km?
In plain words
Some earthquakes happen hundreds of kilometers down, where rock is so hot and compressed that ordinary cracking and frictional sliding should be impossible. Which process lets rock fail suddenly there is disputed.
Precise statement
Earthquakes occur in subducting slabs to about $700\,\mathrm{km}$ depth, where confining pressure exceeds about $1\mathrm{e}11\,\mathrm{dyn}/\mathrm{cm}^{2}$ and frictional sliding would require shear stresses beyond plausible slab stresses. Identify the failure mechanism: transformational faulting in metastable olivine converting to its high-pressure phases, dehydration embrittlement, or thermal shear runaway, and which operates in which slabs and depth ranges. An answer is a mechanism consistent with the depth distribution, focal mechanisms, rupture dimensions and low aftershock productivity of deep events, including large ones such as the 2013 $M_w\,8.3$ Sea of Okhotsk earthquake.
What would settle it
Seismic imaging of metastable olivine wedges matched to deep seismicity, together with high-pressure deformation experiments producing unstable failure at slab conditions.