ILL 1986 Problem 70

Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that $$r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}$$

where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.