The R7 260X can most certainly do it, this is tested here.
On many TV’s you need to turn on “UHD Color” or “Advanced (Color) Setting” manually in the settings of your TV for the HDMI port to which your pc is connected.
Also it could help when connecting 2 screens, to set the screen connected thru the Adapter as second/secondary screen
I strongly advise you to make a custom resolution, set resolution to 3840×2160 at 60Hz and set Timing standard to CVT-RB (reduced blanking);
Open the Crimson Driver software
Click on “screen” ; Then on “additional settings” (right top); Then click “Custom Resolutions” and click on “NEW”. Put in 3480 and 2160 and 60Hz and set Timing Standard to CVT-Reduced Blanking.
O do advise you to try an HDMI cable of which you are certain (certified) it will support 18Gbps or 3840×2160 at 60Hz, perhaps you can borrow one from a friend. Many issue come from cables that are in fact not capable of supporting such high resolutions. Something like this one:
I will send you an email with the latest firmware for the Adapter just in case…