Disjoint-interval conjecture for two-type queue equilibria

Let $(F_a,F_b)$ be a Nash equilibrium arrival profile of the two-type queueing game, with $c3(F_a)$ and $c3(F_b)$ denoting the arrival times used by types $a$ and $b$, respectively. The second type of equilibrium has arrival sets contained in $(0,T]$, where customers of different types arrive on disjoint intervals. Disjoint-interval conjecture. There is no time $t>0$ such that both types of customers arrive simultaneously:

$$c3(F_a)\capc3(F_b)\cap(0,T]=\emptyset.$$

This is a conjectural structural property intended to make the second equilibrium type in the preceding characterization more specific; its resolution is not stated in the source.