Length control for foliations by Jordan curves via smooth quasiconformal maps

Let $\mathrm{\Omega }⊊\mathbb{C}$ be a non-empty simply connected domain and $0<p<2$. Bouchala and Koskela [Proc. Amer. Math. Soc. (2024)] proved that there exists a quasiconformal map $f:\mathbb{D}\to \mathrm{\Omega }$ such that ${\int }_0^1\mathrm{\ell }(f(r{S}^1){)}^{p}dr<\mathrm{\infty }$, and asked whether this conclusion holds at $p=2$, or even for every $0<p<2+\epsilon$ with some $\epsilon >0$. We answer this question affirmatively with $\epsilon =2\mathrm{/}157$ by constructing a single smooth $5$-quasiconformal map $f:\mathbb{D}\to \mathrm{\Omega }$ satisfying this integrability condition simultaneously for every $0<p<2+2\mathrm{/}157$.