Denis's Schanuel conjecture for the Carlitz exponential

Let $A=\mathbb{F}_q[\theta]$, let $K=\mathbb{F}_q(\theta)$, and let $\mathbb{C}_\infty$ be the completed algebraic closure of the completion of $K$ at infinity. Let $u_1,\ldots,u_n$ be elements of $\mathbb{C}_\infty$ that are $A$-linearly independent. Then

$$\operatorname{trdeg}_K K(u_1,\ldots,u_n,\exp_C(u_1),\ldots,\exp_C(u_n))\geq n.$$

Denis's conjecture. The displayed transcendence-degree inequality holds.

This is the Carlitz-exponential analogue of Schanuel's conjecture and is presented as a conjecture of Laurent Denis. Its general status is not resolved in the source.