Bummer, not supported on the Gen2 Learning thermostat yet. I have mine integrated through Homebridge though and it works pretty well. This device called "The Nest Thermostat" is missing a few features the Gen2 has (like schedule learning) but it does look sexier. I installed one for a family member and it works great.
I have to use the Learning Thermostat. None of their more recent ones support my particular system and wiring (electric heat-pump with dual stage heating and cooling). I have it integrated with hubitat thru Google SDM API and so far that works excellent. I am unsure if Matter would even matter to me other than I feel left out anyway.
I must admit, Google killing the WWN API’s happened right before I replaced my Vera hubs with Hubitat. As a result I was forced to switch to a different thermostat.
I haven’t looked into it since - local control via matter world be ideal IMO.