Does solid aluminum oxide conduct oxide ions at high temperatures? Can alumina behave as a solid electrolyte for oxygen ions? Could a cell be constructed with an oxygen cathode and a molten aluminum anode producing more alumina? Would the resulting alumina deposit on the electrolyte? Would it be crystalline?
Alumina is one of the most abundant compounds on Earth. If it would be also a solid electrolyte for oxygen ions, why in the world would anyone go into the trouble of dealing with the whole $\ce{Y2O3/ZrO2}$ thing? No, alumina doesn't and can't behave like that. Its oxygen ions are held in place pretty tightly.

Some forms of alumina may serve as conductors of $\ce{Na+}$ and similar cations, but that's another story.