The conjecture that both bases are transcendental in the integer-power equation

Let $T,R\in\mathbb{R}^+$ satisfy $T^R=R^T=N\in\mathbb{N}$ and $T\neq R$. Suppose $N\neq 16$, so that at least one of $T$ and $R$ is transcendental. Both-bases transcendence conjecture. Both $T$ and $R$ are transcendental. This strengthens the preceding proposition and corollary, which establish only that at least one of the two numbers is transcendental when $N\neq 16$.