ACFT EQUIP PROB TCASII. ACR X FALSE TCASII RA CLB HAD LTSS FROM MLT. PLTDEV.

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WHILE CLBING THROUGH FL250; WBOUND FROM KANSAS CITY; WE WERE CLRED TO FL310 AND TOLD THAT WE WOULD BE CLRED TO OUR REQUESTED FL350 AFTER PASSING EBOUND TFC AT FL330. WHEN MAINTAINING FL310; OVER THE SALINA VOR; I LOOKED AT THE TCASII TO SEE IF I COULD SEE WHEN THE ONCOMING FL330 TFC WAS APCHING; SO THAT WE COULD PLAN OUR CLB TO FL350. THE TCASII WAS SHOWING A TARGET AT OUR 12 O'CLOCK POS WHICH TURNED YELLOW AS A WARNING 'TFC TFC' SOUNDED. UPON CLOSER EXAMINATION; I SAW THAT THE TFC WAS SHOWING FL306. AS I WAS ABOUT TO POINT THIS OUT TO THE OTHER CREW MEMBERS; THE TARGET TURNED RED; AND THE TCASII TOLD US TO 'CLB-CLB;' WHICH I DID IMMEDIATELY; ADVISING ATC OF OUR ACTIONS AT THE SAME TIME. ATC ADVISED US THAT THE ONCOMING TFC WAS AT OUR 1230 O'CLOCK POS AND AT FL330. WE WERE ABLE TO SEE THE TFC; AN MLT; WHICH WAS GOING OFF OUR R WING AT FL330. WE APCHED FL315 BEFORE THE TCASII SAID 'MONITOR VERT SPD' AND 'CLR OF CONFLICT.' WE THEN RETURNED TO FL310. THE TCASII CONTINUED TO SHOW THAT THE TFC WAS AT FL306 UNTIL IT WENT OFF OF THE SCREEN BEHIND US. FAA ACR INSPECTOR FROM THE DEN FAA OFFICE WAS ON OUR JUMPSEAT; AND OBSERVED THIS OCCURRENCE. HE ASKED THAT WE HAVE THE ATC CTLR DETERMINE THE ALTIMETER SETTING OF THE FL330 TFC; AND IF ANY OTHER ACFT WERE IN OUR AREA. WE WERE INFORMED BY ATC THAT THERE WERE NO OTHER ACFT IN OUR AIRSPACE; AND THAT THE MLT WAS SHOWING FL330 ON THE RADAR WITH ITS ALTIMETER SET TO 29.92 AS REQUIRED. SUPPLEMENTAL INFO FROM ACN 234204: I WAS ALERTED BY ACR X THAT HE WAS RECEIVING A TCASII RA. HE STATED THAT THE TFC'S ALT SHOWN FL306 AND HE WAS CLBING TO ANOTHER ALT BECAUSE OF THE TCASII RA. I OBSERVED ACR X MODE C AS FL315. I ISSUED TFC AT 1230 O'CLOCK AT FL330. ACR X SAID HE HAD TFC AT FL330 IN SIGHT AND WAS DSNDING TO FL310. SEPARATION WAS LOST 1500 FT AND 3 MI. THE ONLY TFC IN AREA WAS MLT AT FL330. ACR X RPTED THE POS OF HIS TCASII ACFT AS THE SAME AS MLT AS FL330.