The conjecture that both solutions are transcendental for rational tower bases

Let $Q\in\mathbb{Q}\cap\left(0,e^{-e}\right]$ with $Q\neq 1/16$, and let $h_o(Q)$ and $h_e(Q)$ denote the two solutions identified in the preceding proposition, with $h_o(Q)<h_e(Q)$, of the relevant iterated-exponential equation. The proposition establishes that both values are irrational and that at least one is transcendental. Both-solutions transcendence conjecture. Both $h_o(Q)$ and $h_e(Q)$ are transcendental. This strengthens the second part of the preceding proposition, which proves irrationality of both values and transcendence of at least one; the stronger assertion is presented as a prediction.