I think you are out of luck if the card oes not support the correct HDMI rating.
But I think you could use a $30 UltraAV DisplayPort 1.1 to HDMI 1.4 Active Adapter - AMD Eyefinity Certified to connect to the 4k tv with your HDMI cable and this adapter. Your card is display port 1.2 but it should be backwards compatible and work fine if people are using these adapters to get their macbooks to wook on seiki TV's.
Here is the link to what I am 90%sure will let you do 4k at 30fps.http://www.amazon.com/Accell-B086B-003B ... ewpoints=1
Also I have to say that 6950 has one heck of a cooler on it with those 5 big azz heatpipes and it is very wide also. Also I cannot tell from the pictures but does that card have a bios switch on it? if it does you might be able to flash it up to a 6970 and get some free performance out of that father of the 7950
but I do not know if the 1 GB cards Unlocked....its been a long time.
2600k@4848mhz @1.4v CM Nepton40XL 16gb Ram 2x EVGA GTX770 4gb Classified cards in SLI@1280mhz Stock boost on a GAP67-UD4-B3, SBlaster Z powered by TX-850 PSU pushing a 34" LG 21/9 3440-1440 IPS panel. Pieced together 2.1 sound system