in this argument is that one is led to study the points at which the gradient of the trace attains its maximal, and this makes an important connection between this line of research and that of Open Problem \[16\].

It would be interesting to detect the behavior of a nonlocal minimal graph and of its trace at the corners of the domain and in their vicinity, in particular understanding (dis)continuity and tangency properties, possibly also in relation with the convexity or concavity of the corner. This is related to the analysis of nonlocal minimal cones in either convex or concave sectors with zero exterior datum.

As a first step towards it, one can try to understand how to complete Figure \[DOMA-2\] near $x_1=\pm2$.

Organization of the paper
-------------------------

The rest of this manuscript is organized as follows. Theorem \[FLFL\] is proved in Section \[ISsta\]. The arguments used will exploit a method that we have recently introduced in [@MR3626547] to show that “all functions are locally fractional harmonic”, and a careful discussion of the homogeneous solutions of fractional equations on cones, see [@MR2075671; @MR3810469].

Then, in Section \[15kdttd3\] we present the proof of Theorem \[QC\]. The arguments used here exploit and 