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You can install and play House Flipper on your Linux PC if you enforce the option through House Flipper's Steam properties.
The option you're looking for is called "Force the use of a specific Steam Play compatibility tool" (at the bottom of the properties window).
If the option is missing
First make sure that Steam Play is enabled in settings. Verify settings in the "Steam Play" tab.
If that doesn't help check you can try enabling beta participation in the "Account" tab.