Adiceam–Nesharim–Lunnon's paperfolding counterexample conjecture

Let $\mathcal{\Upsilon}(t)$ be the Laurent series associated with the Paperfolding sequence, and let $t$-$LC$ denote the $p(t)$-adic Littlewood Conjecture with $p(t)=t$. Adiceam–Nesharim–Lunnon's conjecture. If $p\equiv 3\pmod 4$ is prime, then $\mathcal{\Upsilon}(t)$ is a counterexample to $t$-$LC$ over fields of characteristic $p$.

The conjecture is supported by computer evidence, while the source states that the characteristic-$3$ case was proved and this paper supplies further evidence in characteristics $7$ and $11$.