In editing a new Wikipedia article on Rota's basis conjecture, I came across a connection to a different problem in discrete geometry that seems closely related.

When applied to the Euclidean plane (a case for which it is known to be true), Rota's conjecture states the following: suppose you have nine points colored with three colors in such a way that each color forms a non-degenerate triangle. Then the points can be regrouped into three non-degenerate rainbow triangles (triangles with all three colors as vertices). Like so:

Nine points from three monochromatic triangles can always be regrouped to form three rainbow triangles

A different conjecture of Bárány and Larman can also be applied to points in the plane (where again it is known to be true; the harder part is in higher dimensions). For nine points colored in the same way, it states that the points can be regrouped into three rainbow triangles (this time allowed to be degenerate) whose intersection is nonempty.

So it occurred to me to wonder: suppose you have three non-degenerate triangles of three colors, the same as in Rota's conjecture. Do there always exist three rainbow triangles that are both non-degenerate and intersecting? Or maybe stronger, three rainbow triangles whose intersection is full-dimensional, as it is in the figure? And what about higher dimensions (where in general neither conjecture has been proven)?

ETA Feb. 5: There do not always exist three rainbow triangles with full-dimensional intersection; the figure below is a counterexample. At most two of the triangles can cover any point away from the origin, so if three triangles intersect they can only do so at the origin.

Nine points from three monochromatic triangles that cannot be regrouped to rainbow triangles with full-dimensional intersection