ACR IN DSCNT HAS A CONFLICT WITH ANOTHER ACR RESULTING IN LTSS.

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THE PLANES INVOLVED ACR X; AN MLG; DAL BTR A5 MSY AT FL330; AND ACR Y; AN MLG; PBI MCB AEX S5 DFW AT FL310. ACR X WAS GIVEN DSCNT TO FL240 WITH PLT DISCRETION OUT OF FL370. AFTER A MIN OR SO; I NOTICED THAT ACR Y AND ACR X WERE GOING TO BE TIGHT. SO; I TURNED ACR X TO A HDG OF 090 DEGS. I THEN TRIED TO TURN ACR Y. NO ANSWER. SO; I TRIED AGAIN AND IT WAS AT THIS TIME ACR Y TOLD ME HE WAS RESPONDING TO A TCASII ADVISORY AND THAT THEY WERE DSNDING. I TOLD ACR Y TO MAINTAIN FL310 AND FLY HDG 220 DEGS. IT TOOK A TOTAL OF 36 SECONDS FROM FIRST XMISSION UNTIL HE ROGERED TURN. TCASII DID NOT CAUSE THIS ERROR; BUT IT ENHANCED IT GREATLY. ACR Y DSNDED 1400 FT BECAUSE THE CREW LATER STATED THAT THEIR TCASII SHOWED SOMEONE CLBING AT THEM. IF ANYTHING; ACR Y'S TCASII SHOULD HAVE TOLD THE CREW TO CLB. WITH TCASII THE SEPARATION WAS 500 FT AND 2 1/2 MI. WITH NO TCASII ACTION IT SHOULD HAVE BEEN 1900 FT AND 2 1/2 MI. SUPPLEMENTAL INFO FROM ACN 258001: WE WERE FLYING AT FL310 (LEVEL) 30 DME E OF AEX ON J-50 WHEN WE GOT A TCASII ALERT AT 12 O'CLOCK. A FEW SECONDS LATER; WE GOT AN RA COMMANDING A 2500 FPM DSCNT. WE BEGAN OUR DSCNT AND NOTIFIED ATC. AT THIS TIME; ZHU TOLD US TO MAINTAIN FL310 AND TURN TO HDG 220 DEGS (ABOUT A 70 DEG L TURN). THE TCASII WAS STILL SAYING DSND; SO WE KEPT DSNDING AND TURNED TO 220 DEGS. AT ABOUT 29500 FT; I SAW THE OTHER PLANE BELOW US AND TO THE R. WE THEN STOPPED OUR DSCNT AND CLBED BACK TO FL310. ZHU NEVER QUESTIONED US ABOUT LEAVING FL310 AND SOON TOLD US TO FLY DIRECT AEX. I THEN ASKED THEM WHAT HAPPENED AND ALL THEY SAID WAS ANOTHER PLANE (ACR X; I THINK) WAS DSNDING FROM FL330 TO FL290. THEY SAID THEY WOULD NOTE THAT WE DEPARTED OUR ASSIGNED ALT BECAUSE OF A TCASII RA.