Triple-junction singularity conjecture for multi-arc integral equations

Consider a triple junction $4\Gamma$ formed by the intersection of three line segments $4\Gamma_1$, $4\Gamma_2$, and $4\Gamma_3$, whose common point is $4\bm{0} \in {\mathbb R}^2$. For $i,j \in \{1,2,3\}$ with $i \neq j$, let $4\theta_{i,j}$ be the angle between $4\Gamma_i$ and $4\Gamma_j$. If $f$ is the restriction to $4\Gamma$ of a smooth function, let $4\lambda$ solve

$$\mathcal{V}\lambda = f.$$

Triple-junction singularity conjecture. Along $4\Gamma_i$, the solution has a leading-order singularity of the form

$$t^{\min_{j\neq i}\left( \frac{\pi}{\theta_{i,j}}\right)-1}, \quad i=1,2,3,$$

where $t$ is the distance to the branch point $4\bm{0}$. This predicts that the angle between each arc and the other two arcs determines the leading singular exponent of the integral-equation solution at a triple junction. The conjecture is formulated from numerical observations in the paper, and no proof or resolution is supplied.