The non-algebraic-basis conjecture for pi

Let $E\!L^{alg}$ be the field of algebraically based numbers, obtained from $\mathbb{Q}$ by repeatedly adjoining algebraic numbers, exponentials, and logarithms of previously generated numbers. Non-algebraic-basis conjecture.

$$\pi\notin E\!L^{alg}.$$

This conjecture would imply that the circle cannot be squared using straightedge, compass, and anglesectors. The source indicates that it follows from a stronger assumption, the Schanuel conjecture.