The CPU output display data through some Intel proprietary high speed links to the PCH (like H77), then the PCH converts these signals to standard interfaces e.g. VGA/HDMI/DP. Generally the Intel proprietary high speed links is not the bottle neck, so the problem is mostly depend on the mother board spec.
According to https://www.intel.com/content/dam/www/public/us/en/documents/datasheets/7-series-chipset-pch-datasheet.pdf
Figure 5-17 in page 274:
How is display signal output
Each digital port is capable of driving resolutions up to 2560x1600 at 60 Hz through DisplayPort and 1920x1200 at 60 Hz using HDMI or DVI (with reduced blanking).
The PCH support HDMI output directly, there is no need for external chips like DisplayPort to HDMI converter chips. But the built in HDMI doesn't support 4K30Hz at all. So many (may be most?) mother board will use the built in HDMI function to get low cost, for these mother boards, the system will never successfully output HDMI 4K30Hz, don't even try anything which only waste your time. If the limitation is a real hardware limitation, then even you get rid of it with some trick, the performance won't be guaranteed. If the limitation is due to commercial considerations, it will not easy to be get rid off (mostly controlled by some firmware/ROM in the chip instead of kernel/driver).
You can only output HDMI 4K30Hz on mother boards that use an external DisplayPort to HDMI converter chips which support at least 4K30Hz signal rate.
So maybe the best idea is to check the spec of your mother board. If it doesn't declare to support 4K30Hz, don't try it.
About HDMI 1.4: even the mother board declare to support "HDMI 1.4", it may not support 4K30Hz. Since according to Wikipedia HDMI page:
Individual products may have heavier limitations than those listed below, since HDMI devices are not required to support the maximum bandwidth of the HDMI version that they implement. Therefore, it is not guaranteed that a display will support the refresh rates listed in this table, even if the display has the required HDMI version.
My environment: i7-3770(with HD4000 GPU), MSI ZH77A-G43, Windows 10 x64 1903(18362.239).
Product page of ZH77A-G43 says it supports HDMI 1.4, but the customer service says it only support 1920120060Hz, not 4K.