(see Theorem 1.2 in [@2019arXiv190405393D]). Instead, when $n=2$, a similar result does not hold: as a matter of fact, when $n=2$,

-   Theorem \[NLMS\] guarantees that boundary points which attain the flat exterior datum in a continuous way have necessarily horizontal tangency,

-   conversely, boundary points in which the $s$-minimal graph experience a jump have necessarily a vertical tangency (see [@MR3532394]).

Consequently, [*points with vertical tangency accumulate to zero crossing points possessing horizontal tangency*]{}, preventing a differentiable boundary regularity of the surface in a neighborhood of the latter type of points.

The proof of Theorem \[NLMS\] will require a fine understanding of fractional minimal homogeneous graphs. Indeed, as a pivotal step towards the proof of Theorem \[NLMS\], we establish the surprising feature that [*if a homogeneous fractional locally minimal graph vanishes in $\{x_n<0\}$ and it is continuous at the origin, then it necessarily vanishes at all points of $\{x_n=0\}$*]{}. The precise result that we obtain is the following:

\[QC\] Let $u$ be an $s$-minimal graph in ${\ensuremath{\mathbb{R}}}\times(0,+\infty)$. Assume that $$\label{ADX}
{\mbox{$u(x_1,x_2)=0$ if~$x_2<0$.}}$$ Assume also that $u$ is positively homogeneous of degree $1$, i.e. $$\label{ADX6}
u(tx)=tu(x)\qquad{\mbox{ for all $x\in{\ensuremath{\mathbb{R}}}^2$ and~$t>0$.}}$$ Suppose that $$\label{ADX3}
u(0):=\lim_{x\to0} u(x)=0.$$ Then $u(x)=0$ for all $x\in{\ensuremath{\mathbb{R}}}^2$.

We 