The $p$-adic Schanuel conjecture

Suppose $z_1,\ldots,z_n\in\mathbb{D}_p\subset\mathbb{C}_p$ are $\mathbb{Q}$-linearly independent, where $\mathbb{D}_p$ is the domain of the $p$-adic exponential function. Define $\operatorname{tr.deg.}_\mathbb{Q}$ to be transcendence degree over $\mathbb{Q}$. $p$-adic Schanuel conjecture.

$$\operatorname{tr.deg.}_\mathbb{Q} \mathbb{Q}(z_1,\ldots,z_n,\exp(z_1),\ldots,\exp(z_n))\geq n.$$

This is the $p$-adic analogue of Schanuel's classical conjecture and concerns algebraic independence of $p$-adic exponentials. The source gives no resolution, so the conjecture is recorded as open.