Brownawell–Yu conjecture on algebraic independence of Drinfeld logarithms

Let $\rho$ be a Drinfeld $\mathbb{F}_q[t]$-module defined over $\overline{k}$. Suppose $F_{\delta_1},\dots,F_{\delta_s}$ are quasi-periodic functions for $\mathbb{C}_\infty$-linearly independent biderivations $\delta_1,\dots,\delta_s$ in $D(\rho)/D_i(\rho)$, defined over $\overline{k}$. Let $\lambda_1,\dots,\lambda_m\in\mathbb{C}_\infty$ satisfy $\exp_\rho(\lambda_i)\in\overline{k}$ for $i=1,\dots,m$, and suppose that $\lambda_1,\dots,\lambda_m$ are linearly independent over $K_\rho$. Brownawell–Yu conjecture. The $m(s+1)$ quantities

$$\lambda_1,\dots,\lambda_m,\quad \bigcup_{j=1}^s\{F_{\delta_j}(\lambda_1),\dots,F_{\delta_j}(\lambda_m)\}$$

are algebraically independent over $\overline{k}$. This conjecture predicts that, for Drinfeld logarithms of algebraic functions, the linear relations over the multiplication ring of the Drinfeld module are the only algebraic relations. It strengthens the known linear-independence theorem of Brownawell and Yu, while its general status is unresolved.