Weak Ax–Schanuel conjecture for the Gamma function

Let $f_1,\ldots,f_n$ be holomorphic functions satisfying, for every distinct $i,j\in\{1,\ldots,n\}$, $f_i\notin\mathbb{Q}f_j+\mathbb{C}$. Write $\mathbf{f}=(f_1,\ldots,f_n)$ and $\Gamma(\mathbf{f})=(\Gamma(f_1),\ldots,\Gamma(f_n))$. Weak Ax–Schanuel conjecture for the Gamma function.

$$\mathrm{tr.deg.}_{\mathbb{C}}\mathbb{C}(\mathbf{f},\Gamma(\mathbf{f}))\geq n+1.$$

The source describes this functional conjecture as potentially more tractable than the preceding value-transcendence conjectures, and states that it remains open.