Can't use a controller with Steam In-Home Streaming

I'm streaming from my Windows PC to my Linux laptop and trying to use a wireless PS3 controller. The controller works fine when playing games running on the laptop but doesn't get recognized when using In-Home Streaming.

The controller works with games that I've tried to stream (Tomb Raider, Mad Max, etc.) when I run them under Linux, so it's not a problem with the games not supporting it.

Is there some confusion going on between my two computers while streaming? Do I have to pair the controller to the streaming PC and not the laptop for it to be recognized as an input device? Or, maybe, it's not possible to use a PS3 controller on the laptop because the streaming PC doesn't recognize the input?

From the official support article on in-home streaming:

My game controller input doesn't affect the game

If your game controller isn't recognized by any game, check to see if Steam Big Picture recognizes it in the controller settings on the client computer. You may need to configure your controller there in order to use it for streaming. Some games use raw input instead of XInput or DirectInput and are not yet supported.