TL;DR:
Possibly, but most likely not. It also depends on what your laptop's HDMI port supports.
Longer version:
For the common resolution&refresh rate combos, these are your HDMI limitations:
HDMI 2.0:
4k @60Hz
1440p @120Hz
HDMI 1.4:
4k
@30Hz (or maybe only 24Hz?)
1440p @60Hz
HDMI 1.3:
1440p @60Hz
HDMI 1.0-1.2:
1080p @60Hz
There are less common modes supported, as well, but I listed only those that are of concern in this case.
So:
1) If the DGM doesn't support 1440p, then it's probably HDMI 1.2.
2) It is very unlikely that your laptop has HDMI 2.0, so your 4k screen was probably operating at 24/30Hz. But the fact that it displayed a 4k resolution points to the fact that it's at least 1.4, so you should be safe in this regard, in any case.
3) The adaptor you linked to states a bandwidth of 10.2 Gbps and 4k support. That also sounds like HDMI 1.4.
4) Not sure if that adaptor works in both directions, though. It only lists DP as the input, and HDMI as the output. Even the description only gives the example situations as DP-->HDMI.
Because of the fourth point, it probably won't work. Nevertheless, for £10, it might be worth a try. But I wouldn't get my hopes up, if I were you.
Ps. Someone can correct me if I'm wrong, but that's the ballpark, at least.
