Due my research about singularities the problem formulated in this blog post was solved negatively with help of Alex Ravsky who has found a counter-example.
The conjecture was: $latex \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta)$ is closed under finite intersections.
The counter-example follows: $latex f=\{(x,y)\in\mathbb R^2:|x|\le |y| \vee y=0\}$, $latex g=\{(x,y)\in\mathbb R^2:|x|\ge |y| \vee x=0\}$.
It is easy to show that $latex f,g\in \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta)$ but $latex f\cap g\notin \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta)$.
This result is discouraging, because (as it seems) it probably follows that using plain funcoids approach to singularity theory fails and we need to invent something more sophisticated.
Also note that $latex f,g\in \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta)$ is a rather counter-intuitive result (draw a graph to see it), despite of the fact that it can be easily proved.