I can almost guarantee you that if valve doesnt get it working, someone will via moonlight, ALVR, and/or WiVRn.

The amount of work that has gone into the Monado/XRizer/Envision/WayVR stack is phenomenal and gaining momentum.

Last year’s announcement trailer for Valve‘s new hardware mentioned that Steam Machine would be a great option for streaming games to Steam Frame. So it‘s pretty safe to assume that Linux support for streaming is coming.

Given how much valve has been advertising running the steam frame with the steam machine if it doesn't already work now it will be a priority to make it work. I trust valve's fairly consistent marketing more than a leak

Information seems to be mixed, but it's looking like it at least works in some cases: https://www.reddit.com/r/virtualreality_linux/comments/1welkhs/steam_frame_confirmed_by_linustechtips_to_not/
I think what's in question is whether the wifi dongle is widely supported. It might require some manual configuration at first.

WiVRn works in Fedora (over flatpak even!) since March this year, and performs better than SteamVR and possibly even Virtual Desktop - on a Quest 2. WayVR provides desktop portal and playspace dragging. Very recent update makes the interface look good and also provides USB Ethernet support for Android headsets (Quest etc.) if you prefer wired

They recently updated it, which seems to have at least improved stability for me. Still can't get room configuration to run, but I can use default settings fine

Their quick start guide confirms that it does stream from the Steam Machine.

Steam Frame includes a wireless adapater to provide the best quality streaming from your Steam Machine or VR-compatible PC.

Steam Frame: Official Overview and Quick Start Guide (@ 3:40)

Personally I am slightly concerned about their confidence in its performance with the Steam Machine if it wasn't working by the time reviewers got their hands on it.

As of now it does not work with the Steam Machine, which is kind of embarrassing. Though I expect them to fix that ASAP. Hopefully at least a beta version is available when customers get it.

steamos is a linux distro so presumably already working if its supposed to be able to stream from the steam machine. other distros will likely be within a couple days tops if it doesn't work out of the box.

I recall it comes with a USB dongle for 6GHz Wifi streaming. That's most likely going to be supported on Linux

I've streamed Steam games from my Linux PC to my Steam Deck. Other than increased resolution and the foveated streaming, it should be the same or similar enough technology to go to the Frame. I would be surprised if it didn't work. Especially because this "generation" of Steam hardware so far has great interoperability between the Steam Machine and Controller, I would expect them to have thoroughly tested streaming from the Machine to the Frame.