On a problem by Dol'nikov

In 2011 at an Oberwolfach workshop in Discrete Geometry, V. Dol'nikov posed the following problem. Consider three non-empty families of translates of a convex compact set K in the plane. Suppose that every two translates from different families have a point of intersection. Is it always true that one of the families can be pierced by a set of three points?A result by R.N. Karasev from 2000 gives, in fact, an affirmative answer to the “monochromatic” version of the problem above. That is, if all the three families in the problem coincide. In the present paper we solve Dol'nikov's problem positively if K is either centrally symmetric or a triangle, and show that the conclusion can be strengthened if K is a Euclidean disk. We also confirm the conjecture if we are given four families satisfying the conditions above.