SHARYGIN 2015 Problem 23

A tetrahedron $ABCD$ is given. The incircles of triangles $ABC$ and $ABD$ with centers $O_1, O_2$ , touch $AB$ at points $T_1, T_2$ . The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$ . The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.