Optimal integrability conjecture for extension distortions and their inverses

Let $\mathcal G_s(K)$ be the class defined in the paper by, and let $\mathcal F_s(g)$ be the class defined by. For a map $f$, write $K_f$ and $K_{f^{-1}}$ for the distortions of $f$ and its inverse, respectively. Optimal integrability conjecture. The following equations should hold:

$$\inf_{g \in \mathcal G_s(K)} \sup \{q \in (0,\infty): f \in \mathcal F_s(g),\ K_f \in L^q_{\operatorname{loc}}(\mathbb R^2)\}=\max\left\{\frac{1}{Ks},1\right\},$$ $$\sup_{g \in \mathcal G_s(K)} \sup \{q \in (0,\infty): f \in \mathcal F_s(g),\ K_f \in L^q_{\operatorname{loc}}(\mathbb R^2)\}=\max\left\{\frac{K}{s},1\right\},$$

and, for every $g\in\mathcal G_s(K)$,

$$\sup \{q \in (0,\infty): f \in \mathcal F_s(g),\ K_{f^{-1}}\in L^q_{\operatorname{loc}}(\mathbb R^2)\}=\frac{2+s}{s}.$$

These equations concern the conjectured optimal local integrability of the distortion of extensions and their inverses; the paper presents this as the missing part of its theorem, and no resolution is supplied here.