ZKC ASSOCIATE CTLR DESCRIBED OPERROR A FL450 WHEN RADAR CTLR FAILED TO RECOGNIZE DEVELOPING LOSS OF SEPARATION.

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AT XA00Z; I WAS INSTRUCTED TO GO TO SECTOR 31D. I INFORMED THE SUPVR I WOULD DO SO AT MY SHIFT START TIME OF XA06Z. WHEN I SAT DOWN AT THE SECTOR; THE R CTLR WAS VERY BUSY WITH SEVERAL PENDING SITUATIONS. I RECEIVED A SHORT BRIEFING; AND TURNED MY FOCUS TO A NEGATIVE RVSM ACFT'S RTE. ACFT Y (LJ45) WAS ESEBOUND AT FL390 REQUESTING FL450. THE R-SIDE ISSUED A CLB TO FL450. THE URET INDICATED A POSSIBLE CONFLICTION BTWN ACFT Y AND ANOTHER ACFT. I QUESTIONED THE R CTLR AND HE ISSUED VECTORS TO BOTH ACFT AS A RESULT. HOWEVER; BY TURNING ACFT Y TO THE N; ANOTHER CONFLICTION AROSE WITH ACFT X (GLF4) ALREADY AT FL450. THE R-SIDE ATTEMPTED TO TURN ACFT Y BACK R AND DSND ACFT X BUT NOT BEFORE SEPARATION WAS LOST. BOTH ACFT CALLED VISUAL AND INDICATED TFC WAS NO FACTOR. THE RADAR CTLR INDICATED THAT HE WAS NOT FEELING WELL AND HAD WORKED SEVERAL OVERTIME SHIFTS IN THE LAST 3 WKS. THIS; COMPLICATED BY A BUSY SECTOR THAT NEEDED A D-SIDE PRIOR TO MY ARR HAS A SIGNIFICANT BEARING IN MY MIND AS A CONTRIBUTING FACTOR. THAT CAN BE SEEN BY THE OVERSIGHT OF TFC THAT CAUSED THIS ERROR. THE MAIN FACTOR WAS PURE OVERSIGHT OF ALL TFC.