ACR Y RESPONSE TO WRONG CALL SIGN UNAUTH CLB THROUGH OCCUPIED ALT HAD LTSS FROM ACR Z. PLTDEV. SYS ERROR.

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I WAS THE CTLR WORKING EBOUND DEPS FROM SLC. I WAS WORKING ZLC SECTOR 3. 2 ACFT; ACR X AND ACR Y; BOTH STOPPED FOR TFC AT FL270. ACR Z WAS WBOUND AT FL280. I ASKED ACR X HOW LONG BEFORE HE COULD REACH FL290. THE REPLY WAS 'WE CAN DO THAT IN 2 1/2 MINS; ACR Y.' I THOUGHT ACR X HAD REPLIED; SO I SAID 'ACR X; ROGER; CLB AND MAINTAIN FL290; I NEED YOU THERE IN 3 MINS OR LESS.' THE REPLY WAS 'ROGER; CLBING TO FL290; YEAH WE CAN MAKE THAT SPD 360 KTS.' I THEN ISSUED TFC TO ACR Y; AS HIS MODE C INDICATED LEVEL AT FL270.' ACR Y TFC 12 O'CLOCK; 1-5 MI; OPPOSITE DIRECTION B-737; FL280. RPT PASSING THE TFC.' ACR Y REPLIED 'WE'RE IMC.' THE NEXT MODE C UPDATE SHOWED ACR Y FL276 CLBING. I ASKED HIS ALT; HE SAID FL283. CLOSEST PROX WAS 1.4 NM; 600 FT. SUPPLEMENTAL INFO FROM ACN 303970: ATC ISSUED ACR X A CLRNC TO CLB TO AND MAINTAIN FL290 WITHIN 3 MINS OR LESS. THE CLRNC WAS ACKNOWLEDGED AND THE CLB TO FL290 WAS STARTED. THE TCASII WAS ON THE 14 NM RANGE AND NO TARGETS WERE DISPLAYED. AT APPROX FL278; A TARGET APPEARED ON THE TCASII AT 11:30 O'CLOCK POS; 13 NM. THIS WAS FOLLOWED BY A TA. ACR X INITIATED A 10 DEG TURN TO THE R; AND INCREASED THE RATE OF CLB. THE ATC CTLR ASKED US TO VERIFY THAT WE WERE CLRED TO REMAIN AT FL270. ACR X REPLIED 'NO;' THAT WE WERE CLRED TO FL290; AND THAT WE HAD INITIATED A TURN TO THE R FOR TFC. THE ATC CTLR THEN ATTEMPTED TO TURN ACR X BACK TO THE L; WHICH WOULD HAVE CROSSED ACR X IN FRONT OF THE ONCOMING TFC. AT THAT TIME THE TCASII ISSUED AN RA. THE TFC WAS AT APPROX 11 O'CLOCK POS AND CLOSING WITHIN APPROX 1 MI. ACR X CONTINUED A R TURN TO APPROX 40 DEGS OFF OF THE ORIGINAL HDG AND INCREASED THE RATE OF CLB TO 2000 FPM. X LEVELED AT FL290 IN 2 MINS. THE OTHER ACFT WAS NEVER GIVEN ANY HDG CHANGE. THE TFC PASSED OFF THE L AT APPROX 11 O'CLOCK AND ABOUT 500 FT BELOW US. THE TFC WAS APPROX 1/2 MI AWAY FROM ACR X. SUPPLEMENTAL INFO FROM ACN 303971: WHILE NAVING ON J173 SLC-EKR; MAINTAINING FL270; ZLC GAVE OUR FLT; ACR X; A CLRNC TO CLB SO AS TO BE LEVEL FL290 IN 3 MINS OR LESS. WE DEPARTED FL270 FOR FL290; AT WHICH TIME A TARGET APPEARED ON OUR TCASII. THE TARGET WAS 12 O'CLOCK; 280 DEGS; HDG W. WE TURNED 40 DEGS TO THE S TO EVADE THE ONCOMING TARGET. THE ZLC CTLR THEN ASKED US TO VERIFY THAT WE WERE MAINTAINING FL270. WE IGNORED THE HDG CLRNC AND REMINDED THE CTLR OF OUR CLRNC TO FL290 WITHIN 3 MINS OR LESS. WE LEVELED AT FL290 WITHIN 2 MINS. SUPPLEMENTAL INFO FROM ACN 304261: ATC ISSUED OUR FLT A CLB CLRNC FROM FL270-FL290 IN 3 MINS OR LESS. WE READ BACK THE CLRNC AND INITIATED CLB. ATC CTLR SEEMED TO BE NERVOUS AND CONFUSED THROUGHOUT THE EVENT. ATC ISSUED NO INSTRUCTIONS TO THE OTHER ACFT AT ANY TIME NOR DID HE EVER TELL US ABOUT THE TFC AT ANY TIME.