Vertex conjecture for subdifferentials of bivariate rational functions

Let $r$ be the rational function defined in the paper, let $P$ be a polytope, let $f(x)=r(x)+I_P(x)$, and let $v$ be a vertex of $P$ satisfying $\xi_1(v)=\xi_2(v)=0$. Vertex conjecture. The subdifferential $\partial f(v)$ is a parabolic region. This concerns the exceptional vertex where both the numerator and denominator vanish, although the rational function extends continuously over the polytope; the claim is conjectured from numerous observations.