for the sake of generality, some results are proved in arbitrary dimension $n{\geqslant}2$, whenever the proof would not experience significant simplifications in the case $n=2$ (then, for the proof of Theorem \[QC\], we restrict ourselves to the case $n=2$, see also Open Problem \[17\]). As customary, given $E\subset{\ensuremath{\mathbb{R}}}^{n+1}$, it is convenient to consider the nonlocal mean curvature at a point $x\in\partial E$, defined by $$\label{jfgnjbj96768769} {\mathcal{H}}_E^s(x):=\int_{{\ensuremath{\mathbb{R}}}^{n+1}}
\frac{\chi_{{\ensuremath{\mathbb{R}}}^{n+1}\setminus E}(y)-\chi_E(y)}{|x-y|^{n+1+s}}\,dy.$$ The first step to prove Theorem \[QC\] is to establish the existence of a small vertical cone not intersecting the boundary of a homogeneous nonlocal minimal surface on a hyperplane with null exterior datum. Letting $e_{n+1}:=(0,\dots,0,1)\in{\ensuremath{\mathbb{R}}}^{n+1}$, the precise result that we have is the following one:

\[3243-237gndlsm\] Let $E\subset{\ensuremath{\mathbb{R}}}^{n+1}$ be a locally $s$-minimal set in ${\ensuremath{\mathbb{R}}}^n\times(0,+\infty)$. Assume that $$\label{ADX-EE}
E\cap\{x_n<0\}=\{x_{n+1}<0\}\cap\{x_n<0\}$$ and that $$\label{ADX6-EE}
{\mbox{$tE=E$ for every~$t>0$.}}$$ Then, $$\label{6BOu-1BIS}
\min\Big\{ {\rm dist}\big(e_{n+1}, (\partial E)\cap\{x_n>0\}\big),\;
{\rm dist}\big(-e_{n+1},(\partial E)\cap\{x_n>0\}\big)\Big\}>0.$$

We claim that $$\label{6BOu-1} e_{n+1}\not\in
\overline{(\partial E)\cap\{x_n>0\}}.$$ Indeed, suppose by contradiction that $e_{n+1}\in\overline{(
\partial E)\cap\{x_n>0\}}$, and thus, by , also $te_{n+1}\in\overline{(\partial E)
\cap\{x_n>0\}}$, for all $t>0$. By , we have that $B_{1/2}(e_{n+1})\cap\{x_n<0\}\subset {\ensuremath{\mathbb{R}}}^{n+1}\setminus E$. Hence (see e.g. Theorem B.9 in [@MR3926519]), we have that ${\mathcal{H}}_{E}^s(te_{n+1})=0$ for all $t\in\left[\frac{9}{10},\frac{11}{10}\right]$. In particular, if $F:=E-\frac{e_{n+1}}{100}$, we see that $F\subset 