Thanks for the responses. I bet conduction away by materials is the problem, among others.
Vaporization of the material on the inside of the cavity would spike the pressure... I would think this would keep the temperature up but 30 million deg kelvin may be too high a bar.
Usually people talk about magnetic confinement schemes or some such thing for fusion reactions presumably because the pressure and temperatures would vaporize/destroy all conceived material vessels too quickly. I would like to hear someone talk about the limits of materials in this realm. So...