Stronger transcendence conclusion for equal exponential powers

Let $r,s,t$ be as in the preceding proposition: $r,s,t$ are positive real numbers with $s<t$ and $s^{\,t}=t^{\,s}=r$, where $r>e^{\,e}$. Under either of the following conditions, at least one of $s$ and $t$ is known to be transcendental: (i) $16\neq r\in\mathbb{N}$, or (ii) $r^n\in\overline{\mathbb{Q}}\setminus\mathbb{Q}$ for every $n\in\mathbb{N}$.

Stronger transcendence conclusion. Under the hypotheses of Proposition $(i)RT=TR=A$, both $s$ and $t$ are transcendental.

This prediction strengthens the proposition's unconditional conclusion from at least one transcendental number to both numbers being transcendental. The source attributes it to an earlier conjecture and gives no resolution.