The general $v$-adic convergence conjecture for Hayes-module exponentials and logarithms

Let $\rho$ be a Hayes module on $X$, with no restriction on the genus of $X$ or on $d_\infty$, and let $H^+$ be the field of definition of $\rho$. Fix a place $v$ of $X$ different from $\infty$, normalized so that the value group is $\mathbb{Z}$. Let $K_V$ be the finite field extension of $H^+$ containing all zeros of the Drinfeld divisor $V$ corresponding to $\rho$. Fix an embedding $\overline K\to\overline K_v$, let $w$ be the place of $K_V$ over $v$, normalized so that its value group is $\mathbb{Z}$, and let $e_w$ be the ramification index of $w$ over $v$.

General $v$-adic convergence conjecture. The exponential series $e_\rho(z)$ converges in $\mathbb{C}_v$ whenever

$$w(z)>w(J_0)+e_w\cdot e_\theta\cdot\frac{1}{q^{\frac{\deg \mathfrak{p}}{f_\theta}}-1},$$

where $J_0$ is an ideal depending on the Drinfeld divisor $V$ evaluated at $\Xi$, and $e_\theta,f_\theta$ are positive integers depending respectively on ramification and inertia. The logarithm series $\log_\rho(z)$ converges in $\mathbb{C}_v$ whenever $v(z)>0$, and the $v$-adic valuation of its coefficients has order of magnitude $O(n)$.

This conjecture extends the convergence results from the genus-zero example to Hayes modules on arbitrary curves and with arbitrary $d_\infty$. A principal difficulty is obtaining an explicit integral model for the shtuka function with all zeros integral and expressing the differential $\omega$ in terms of that shtuka function.