ZID Controller providing OJT described a near loss of separation event when several assumptions were made by the controllers involved and vigilance was not maintained.

Narrative

I was providing on the job training with a R-Associate Developmental. Air Carrier X was climbing eastbound on the RHOMM STAR. ZID 69/24 pointed out the aircraft to us at ZID 26 while the aircraft was out of FL190 climbing to FL230 about 25 west of YRK. Shortly after; ZID 85 began a hand off on Air Carrier Y; east of HNN. Conflict Alert activated between Air Carrier Y and a CMH departures climbing southbound; ZID 85 working both aircraft. The ZID 85 R-Associate called us and requested lower on Air Carrier Y. The Developmental; after conferring with me; approved descending to FL190. At this point; we recalled that Air Carrier X was in hand off mode to ZID 85. The Developmental made it clear that we were releasing only in our airspace; as Air Carrier Y was still over the lateral limits of ZID 69/24. ZID 85 did not descend Air Carrier Y as expected; and did not take the hand off and climb Air Carrier X. About 2 minutes after crossing over the lateral limits of our sector; ZID 85 descended Air Carrier Y and switched the aircraft to our frequency. Air Carrier X was obviously traffic by this time; and we were surprised that ZID 85 would descend right into known traffic. The ZID 26 R-Side expedited Air Carrier Y's descent and achieved the separation at literally the last second. This situation required a series of small events to occur: 1. ZID 69/24 pointed out the aircraft as climbing into the high altitude stratum; that didn't happen. 2. When ZID 26 approved lower on Air Carrier Y; the sector expected for the aircraft to descend right away; instead; the descent began at the worst possible time. 3. ZID 85 did not take the hand off on the climbing Air Carrier X; creating an unexpected conflict. 4. ZID 85 did not see Air Carrier X until conflict alert activated; ZID 26 should have referenced Air Carrier X when the request for lower on Air Carrier Y was made; even though it did not appear to be traffic. ZID5 should have descended the aircraft right away; or call back to ZID 26 and explain the delay. ZID 69/24 should have queried ZID 85 on why they were ignoring the hand off. A very large factor in this situation; and countless others like it; is the design of the RHOMM STAR and GAVNN/BRUSH STARS. The confliction point between the two involves 3-4 sectors. How to fix: Eliminate the HNN transition of the GAVNN and BRUSH STARS. Have the aircraft come in over HVQ. This moves the confliction point so much farther south; giving both climbing and descending aircraft much more time to swap out. It also moves the confliction point well into the lateral confines of ZID 26's airspace; and limiting the number of sectors and situations that will occur.