The $p(t)$-adic Littlewood Conjecture over finite fields

Let $\mathcal{F}_q$ be a finite field, let $\mathcal{F}_q((t^{-1}))$ be the field of formal Laurent series, and let $|\Theta(t)|=q^{\deg(\Theta(t))}$ be the norm. For an irreducible polynomial $p(t)\in\mathbb{F}_q[t]$, define $|N(t)|_{p(t)}=|p(t)|^{-\nu_{p(t)}(N(t))}$. Let $|\langle\cdot\rangle|$ denote the distance to the nearest polynomial in $\mathcal{F}_q[t]$. The $p(t)$-adic Littlewood Conjecture. For every $\mathcal{\Theta}(t)\in\mathbb{F}_q((t^{-1}))$ and every irreducible $p(t)\in\mathbb{F}_q[t]$,

$$\inf_{N(t)\in\mathbb{F}_q[t]\backslash\{0\}}|N(t)|\cdot |N(t)|_{p(t)}\cdot |\langle N(t)\Theta(t)\rangle|=0.$$

The paper studies this function-field analogue over finite fields; the supplied status is unknown, and the introduction describes explicit counterexamples in odd characteristic.