No, you cannot do this in current form. The converter is USB to HDMI, not "HDMI-to-USB". It does not work in reverse. These are two entirely different functions, and the HDMI-to-USB does not exist.
The dongle in your link is a USB device of video class. The host detects this device and works with it as it is a videocard. This "videocard" outputs HDMI format video stream for your PC monitor. It is OUTPUT.
To accomplish what you want, you would need to design a sort of emulation of PC monitor:
(a) make a HDMI receiver;
(b) make a USB device that looks like a video camera, USB webcam class, which uses the HDMI stream and converts it into webcam format;
This it technically doable, but the video lag will be horrible, so gamers will not like it.