Pages 39-41 of the Peterka report state that the Legacy was held at FL310 from 6:05 to 6:10 to allow another plane to pass at FL320. Those levels are within 1000 feet of each other, and as no TA was issued, the TCAS was never turned on.
For two aircraft flying straight and level at FL310 the TA vertical threshold is 850ft. So, if the two aircraft were 1,000ft apart, a TA would not be issued.