Transcendence of solutions to a nested exponential equation

Let $\alpha\neq0$ and $z$ be complex numbers, with $\alpha$ algebraic and $z$ irrational.

Nested exponential transcendence conjecture. If

$$\alpha^{\,\alpha^{\,z}}=z,$$

then $z$ is transcendental.

This is presented as a stronger statement than the consequence of the Gelfond–Schneider theorem for equations of the form $\alpha^{\,z}=z$. The source gives no unconditional proof or resolution.