Carlitz–Schanuel conjecture over Tate algebras

Let $A=\mathbb{F}_q[\theta]$, let $K_s=\mathbb{F}_p(\theta,t_1,\ldots,t_s)$, and let $\mathbb{K}_s$ be the ambient field introduced for the extension of the Carlitz exponential. If $u_1,\ldots,u_n\in\mathbb{K}_s$ are linearly independent over $A[t_1,\ldots,t_s]$, then

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

Carlitz–Schanuel conjecture over Tate algebras. The displayed transcendence-degree inequality holds.

This generalizes Denis's conjecture to the Tate-algebra setting. The source presents it as a conjectural statement, with no resolution supplied.