As always, resolution and Hz is entirely irrelevant to this.
The laptop has no HDMI outputs. It has DP outputs and active adapters may be able to convert this to HDMI. How successfull this works for specific features and monitors and GPUs (and their drivers) is varies greatly.
Depending on the port and generation of your laptop the DP speeds supported are different.
At the very least its HBR3. For the Phoenix AMD board, it might be that the no-USB4 DP Alt mode output supports UHBR10 speeds as well, Frameworks specs are woefully unclear for any of this (the CPU itself supports UHBR10 on every DP output, but the mainboard definitely does not for the USB4 ports, unclear for the rest. The chips used would support it, but if Framework did not test it, they might still justifiably have disabled it in firmware to prevent issues. The new Strix Point AMD board for example should support UHBR10 on every board, but I also think this has not been officially confirmed. While the Intel Ultra 100 board should support UHBR10 and UHBR20 on every port)
All currently available DP HDMI adapters are limited to HBR3 speeds anyway. So the max. bandwidth that they can get from any source is ~25.9 Gbit/s. The current max usable bandwidth of an FRL 6 connection would be 42.6 Gbit/s. While the current crop of adapters can output at max bandwidth, that would be done by using higher compression from the GPU and the adapter decompressing. Which basically no sane HDMI monitor would do / need.
UHBR10 DP connections with 38.8 Gbit/s of usable bandwidth would solve for most of that with future DP-HDMI adapters.
VRR is a problem. Synaptics announced a DP-HDMI converter chip that can translate DP Adaptive Sync to HDMI VRR, but hard to know which adapter uses it and actually can do it.
There have been documented cases of people using one specific Cablematters adapter (fullsize DP) with AMD dGPUs under Linux and have HDMI VRR support (even though the Cablematters product description says it does not have it. There also seem to be 2 variants of that adapter, one with an old chip without it and the new one with it).
But I have basically the USB-C variant of that adapter (same chip - VMM7100) and it does not seem to support it in practice (either on Intel Xe iGPUs nor on an AMD Strix Point iGPU. But also the only monitor with HDMI VRR I have is using HDMI VRR without FRL, which seems rare at best. Intel’s sparse documentation for example says, that they do not support this combination on a driver level at all even with native ports. AMD Strix Point supports HDMI VRR with this monitor via native HDMI port, but does not detect VRR support with said DP-HDMI adapter).
So, you require practical proof of somebody else using a very specific adapter to get HDMI VRR with a specific type of monitor to be sure, as almost no manufacturer of such adapters is currently advertising it (because the software / GPU driver and monitor supports seems spotty at best, they probably don’t dare to advertise it, because there are too many variables to be sure. Many people cannot even distinguish between FreeSync HDMI and HDMI VRR, and only the latter would ever be supported by such an adapter, as the former is proprietary garbage).
Best to just avoid HDMI as much as possible and just stick to DP where this is a lot less of a problem.