My Mac Pro automatically installed the 144.0.0.0.0 firmware when I was using Mojave as my primary OS on my machine at the time. But I seem to recall needing an additional graphics card (in my case the NVIDIA GTX 660 Ti) because the stock 5770 didn't work when I tried to update, but that may have just been me, not sure. I didn't include anything for updating the firmware, because mine was updated back in 2019 when the firmware was launched. So, I would hunt down a guide from someone else to get you that far.
But now having seen other posts about requiring HII, this whole process might not work for you; I'm not sure. But it might give more data points for the community if you do decide to pursue it.
Yes I know, but trying nothing cost better than do nothing I will trying to follow your procedure with your 4070ti
For my experience full Mojave OS works with ATI 5770 so let me know if I can proceed to install mojave compleatly