Transcendence conjecture for power towers of ultra-Liouville numbers

Let $\theta$ be the fixed point of $x\mapsto \xi^x$ and define the iterated power towers by $h_1(\xi)=\xi$ and $h_{k+1}(\xi)=\xi^{h_k(\xi)}$ for $k\geq 1$. An ultra-Liouville number is a Liouville number satisfying the stronger approximation condition used in the paper. Transcendence conjecture. If $\xi$ is an ultra-Liouville number, then $h_k(\xi)$ is transcendental for all $k\geq 1$. The conjecture extends the established transcendence results for the first few levels of the power tower. Corollary (ii) shows that infinitely many of $h_4(\xi),h_5(\xi),\ldots$ are transcendental for $\xi\in\mathbb{L}^{(\infty)}$, but it does not prove transcendence at every level.