ACR X NON ADHERENCE TO ATC CLRNC TCASII TA HAD LTSS FROM ACR Y. SYS ERROR. PLTDEV. EVASIVE ACTION TAKEN.

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ZOA VIA HNL ARINC REQUESTED ACR X CLB FROM FL370 TO FL390. AS WE CLBED; AT APPROX FL385 TURB BEGAN TO GET MODERATE PLUS AND IT WAS OBVIOUS IT WOULD BE UNSAFE TO CONTINUE CLB; ADVISED HNL ARINC UNABLE TO CLB AND WAS DSNDING BACK DOWN TO FL370. AT APPROX FL380 IT WAS APPARENT ANOTHER ACR Y WAS AT FL370 APPROX 7 NM AWAY BY TCASII AND VISUAL SIGHTING. WE LEVELED AT FL380 AND ADVISED ARINC OF CONFLICT AND LEVELED AT FL380. ARINC (ZOA) THEN CLRED BACK DOWN TO FL370; LATER ADVISED BY ARINC TO CONTACT ZOA SUPVR AND DID SO UPON LNDG. ZOA SAID POSSIBLE PLTDEV; BUT CTLR APPARENTLY ALSO ERRED SINCE HE CLRED OTHER ACFT TO FL370 BEFORE WE GOT TO FL390. WE NEVER GOT TO IT OR RPTED LEVEL. SUPPLEMENTAL INFORMATION FROM ACN 259956: ATC REQUESTED ACR X CLB FROM FL370 TO FL390 AND RPT LEVEL. SOME LIGHT CHOP AT FL370. DURING CLB ENCOUNTERED MODERATE TURB AS WE APCHED FL390. TOLD ARINC WE WOULD BE UNABLE TO MAINTAIN FL390 AND WERE RETURNING TO FL370. AS WE DSNDED WE SAW ON OUR TCASII THAT ACR Y THAT HAD BEEN AT FL350 HAD JUST LEVELED AT FL370 APPROX 7 NM AHEAD OF US. WE NOTIFIED ARINC THAT WE WOULD LEVEL OFF AT FL380 AND THAT WE COULD MAINTAIN FL380. ATC CLRED US BACK TO FL350 AND RECLRED US TO FL370.