ACR X TCASII TA DSCNT THROUGH OCCUPIED ALT HAD LTSS FROM ACR Y. SYS ERROR.

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ACR X WAS FL390 NWBOUND. ACR Y WAS FL370 SEBOUND; X WAS ISSUED A DSCNT TO FL310. WHEN THE CONFLICT WAS NOTICED; X WAS AT FL389 AND DSNDING. X WAS TOLD TO CLB AND MAINTAIN FL390 EXPEDITIOUSLY AND TO RPT LEVEL. X DID NOT RESPOND TO THE FIRST CALL. THE CALL WAS REPEATED AND X ACKNOWLEDGED AND RPTED TFC IN SIGHT. X PASSED BEHIND Y BY 3.8 MI AND 1100 FT. THERE WERE APPROX 15 ACFT ON FREQ AND THERE WAS AN ASSISTANT CTLR; RADAR CTLR INSTRUCTOR; AND A HDOF CTLR ON THE POS IN ADDITION TO MYSELF. SUPPLEMENTAL INFO FROM ACN 285382: I HAD JUST PLUGGED IN ON THE HDOF POS AT BAE HIGH SECTOR AND WAS REVIEWING THE FLT DATA WHEN THE RADAR TRAINEE DSNDED ACR X TO FL310 WITH FL370 XING TFC. I IMMEDIATELY CALLED THIS TO HIS ATTN AND HE REISSUED AN ALT OF FL390 TO THE ACFT THAT HE JUST DSNDED. THE ACFT CLBED BACK TO FL390; BUT NOT BEFORE WE LOST SEPARATION WITH ACR Y XING TFC. SUPPLEMENTAL INFO FROM ACN 285372: ON-THE-JOB-TRAINING WAS BEING CONDUCTED. I WAS DISTRACTED BRIEFING HDOF CTLR WHO HAD JUST PLUGGED IN. TRAINEE DID NOT REALIZE ACFT WERE IN CONFLICT. INSTRUCTOR DID NOT CATCH DSCNT CLRNC UNTIL TOO LATE. CORRECTIVE ACTION WAS INITIATED TOO LATE TO AVOID LOSS OF SEPARATION. SUPPLEMENTAL INFO FROM ACN 285366: ACR X AT CRUISE AT FL390; ZAU GAVE X A CLRNC TO DSND TO FL310. X CALLED OUT OF ALT AND BEGAN DSCNT. AT FL380 GOT A TA ON TCASII 12 O'CLOCK 1000 FT BELOW. AT SAME TIME ZAU TOLD US TO CLB BACK TO FL390; CONFLICTING TFC. OTHER ACFT ASKED ABOUT OUR ALT AND ATC AGAIN STATED HIS FAULT. SUPPLEMENTAL INFO FROM ACN 285811: STARTED DSCNT FROM FL390 TO FL310 AS ASSIGNED BY CTR. PASSING FL385; CTR CALLED BACK AND TOLD US TO CLB TO FL390 FOR TFC. I LOOKED OUT THE WINDOW AND SAW TFC BELOW AND AT 1 O'CLOCK POS. ALMOST IMMEDIATELY A TCASII TA WAS ANNOUNCED 'TFC.' I POINTED OUT TFC TO OTHER CREW MEMBERS AS CLB WAS INITIATED BACK TO FL390.