AN ACR LTT DELAYED ITS DEP FROM AN ALT BECAUSE OF A POSSIBLE TCASII THREAT ADVISORY.

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MY FO WAS FLYING WHILE I WAS HANDLING ATC COMS. WE WERE TCASII EQUIPPED. MY FLT WAS INBOUND TO LA GUARDIA. THE CTLR ISSUED A DSCNT FROM 6000 TO 5000 FT. IMMEDIATELY AFTER WE ACKNOWLEDGED THE DSCNT; THE TCASII GAVE US A TA AT MINUS 500 FT RELATIVE ALT. WE DELAYED OUR DSCNT FOR ABOUT 1 MIN. THE CTLR WAS VERY BUSY AND WE DID HAVE THE OPPORTUNITY TO CALL HIM BACK TO NOTIFY HIM OF OUR DELAYED DSCNT. I ELECTED TO DELAY THE DSCNT TO AVOID AN INTENTIONAL TCASII RA WHICH WOULD HAVE RESULTED IN MY OPINION A CLB THAT WOULD HAVE TAKEN US ABOVE 6000 FT. A CALL FROM THE CTLR LATER IN THE AFTERNOON REVEALED THAT WE; ATC AND PLTS; HAD A LOT TO LEARN ABOUT THIS NEW EQUIP AND EFFECTIVE WAYS TO USE IT SAFELY. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTING PLT DID NOT SEE THE INTRUDING ACFT; BUT HE BELIEVES THAT THERE WAS AN ACFT. HE DOES NOT BELIEVE THAT THIS WAS A FALSE ALARM. THE TRACON CTLR WAS RATHER UPSET ABOUT THE RPTR NOT LEAVING HIS ALT IMMEDIATELY. THE TRACON CTLR TRACKED DOWN THE RPTR BY CALLING THE ACR DISPATCH; THEN CREW SCHEDULE; THEN THE PLT'S LOUNGE AT LGA. THE CTLR REITERATED THAT HE HAD WANTED THE ACFT OUT OF ITS PREVIOUS ALT NOW; NOT WHENEVER THE RPTR GOT AROUND TO IT. THERE HAS BEEN NO FURTHER FAA ACTION.