Denis's strengthened Carlitz–Schanuel conjecture

Let $A=\mathbb{F}_q[\theta]$, let $K=\mathbb{F}_q(\theta)$, and let $p$ be the characteristic of $\mathbb{F}_q$. For $A$-linearly independent elements $u_1,\ldots,u_n\in\mathbb{C}_\infty$, consider the Carlitz exponential and its higher derivatives with respect to $\theta$, denoted by $\exp_C^{(i)}$. Then

$$\operatorname{trdeg}_K K(u_1,\ldots,u_n,\exp_C(u_1),\ldots,\exp_C(u_n),\ldots,\exp_C^{(p-1)}(u_1),\ldots,\exp_C^{(p-1)}(u_n))\geq pn.$$

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

The source describes this as a strengthening proposed by Denis to encompass algebraic-independence results involving derivatives in the parameter $\theta$. Its general status is left unresolved.