Chow–Rimanic function-field Lonely Runner Conjecture

Let $q$ be a prime power. In the Laurent-series field $\mathbb{F}_q((X))$, write $\alpha=[\alpha]+\|\alpha\|$ with $[\alpha]\in\mathbb{F}_q[X]$ and fractional part $\|\alpha\|\in\mathbb{T}$, where

$$\mathbb{T}=\{\alpha\in\mathbb{F}_q((X)): \operatorname{ord}(\alpha)<1\}.$$

For $F\subset\mathbb{F}_q[X]\setminus\{0\}$, define

$$\kappa_q(F)=\sup_{\alpha\in\mathbb{T}}\min_{f\in F}|\alpha f|.$$

Function-field Lonely Runner Conjecture. For every prime power $q$ and every finite set $F\subset\mathbb{F}_q[X]\setminus\{0\}$ of polynomials satisfying

$$1\leq|F|<\frac{q^{k+1}-1}{q-1}\quad\text{for some }k\in\mathbb{N},$$

we have

$$\kappa_q(F)\geq q^{-k}.$$

This is the function-field analogue proposed by Chow and Rimanic. It is collected among the survey's open problems and remains open in the stated generality.