FLC OF ACR LGT ACFT TOOK EVASIVE ACTION IN RESPONSE TO A TCASII RA RESULTING IN PAX INJURY.

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AN LGT AIRLINER TOOK EVASIVE ACTION FOR AN FGT OPERATING VFR IN OFFSHORE AIRSPACE.THE FGT WAS RECEIVING FLT FOLLOWING. THE LGT; A SCHEDULED ACR; WAS IFR AT FL350. A SIT DEVELOPED THAT THE FGT ACFT WAS SBOUND OUT OF FL340 CLBING HEAD-ON IN CONFLICT WITH THE NBOUND LGT. TFC WAS CALLED TO BOTH PLTS. THE FGT PLT WAS ASKED TO STAY AT FL340. THE PLT OF THE LGT REACTED TO A TCASII PULLUP AND CLBED RAPIDLY AND TURNED R. THE PLT OF THE LGT THEN SAID HE SAW THE FGT GO BY. THE PLT OF THE FGT NEVER SAID HE SAW THE LGT. AS A RESULT OF THIS INCIDENT; A PAX ON THE LGT SUFFERED A COMPOUND FRACTURE OF THE ANKLE. THE RADAR SECTOR I WAS CTLING WAS VERY BUSY AND THE TFC WAS COMPLEX. THE FACTORS THAT ALLOWED THIS SIT TO DEVELOP ARE AS FOLLOWS: VFR ACFT OPERATING IN THE HIGH ALT ENVIRONMENT. THE PLT OF THE VFR FGT NOT ABIDING BY THE CTLR'S PREVIOUS SUGGESTION TO MAINTAIN AN ALT ABOVE FL370. ICAO REGS AND OTHER PROCS WHICH PERMIT NASA ACFT TO OPERATE VFR ABOVE FL200 IN OFFSHORE AIRSPACE. SUPPLEMENTAL INFO FROM ACN 235860: I HAD FILED A VFR FLT PLAN; TOOK OFF FROM ELLINGTON AT PM30L; AND PROCEEDED TO OUR FUNCTIONAL CHK FLT (FCF) WORK AREA. HAVING COMPLETED THE HIGH ALT ENG CHKS; I BEGAN A DSCNT WHILE INITIATING THE MAX PERFORMANCE MANEUVERS FOR THE FCF. WHILE IN A LOW AIRSPD MODE; THE CTLR CALLED OUT TFC AT FL350 AND REQUESTED THAT I CLB TO ABOVE FL350 QUICKLY. APPLIED MAX PWR; BUT DUE TO MY LOW ENERGY STATE; WAS UNABLE TO RAPIDLY RESPOND TO HIS REQUEST. SEEING THAT I WAS CLBING SLOWER THAN DESIRED; HE ASKED ME TO LEVEL AT FL340. I QUICKLY DSNDED BACK TO FL340; AND SHORTLY AFTERWARD THE COMMERCIAL ACFT RPTED THAT THEY HAD GOTTEN A PULL-UP COMMAND FROM THE TCASII; AND THEY HAD RESPONDED TO IT. I NEVER ESTABLISHED VISUAL CONTACT WITH THE OTHER ACFT.