If shooting RAW, is the white balance selected in camera irrelevant for exposure?

I have read the existing question - Does the camera white balance setting affect the raw image at all?

After reading the above post, I am still not clear on the part around exposure. Is it possible that selecting one white balance over the other can cause the exposure to be different? This answer from that question in particular brings that point up exactly: https://photo.stackexchange.com/a/3598/4892

As far as I have heard, the WB does not affect the RAW data, but it does affect the exposure.

So in difficult lighting situations the camera auto exposure might react differently depending on the WB.

For an example, what if I always shot in RAW and always shot in Auto WB, or always Incandescent WB. If I am WB correcting in post regardless and ignore any "preview", am I losing any information or altering the exposure by leaving the WB in one of these settings?

I found additional discussion of this topic but no real answer here. I also found someone who recommended instead of Auto WB, to always leave the camera in WB 5000k to keep the most information.

Answer

This is actually fairly easy to test, and so I did. I used my Pentax K-7, so this doesn't speak to all cameras, but I think at least many work the same way.

I worked in a dark room, lit only by an iPad app which simply turns the whole screen a certain color. I put the camera close enough to the screen that the color filled the entire frame, and, although I don't think it matters, manually focused. I put the camera in aperture priority mode with the lens at F/2.8 and ISO fixed at 1600. And I set capture to RAW.

With a red light and daylight white balance, the camera selected shutter speed 800. When I set white balance to tungsten (without changing anything else), it instead chose 640. I went back and forth several times to make sure nothing else was influencing the result.

Then I changed the light to blue. Here, with tungsten, the camera picked 1250, but with daylight, it chose 1000.

So, clearly the white balance selected does influence metering decisions made by the camera even in RAW mode.

However, it's also worth noting that even in this contrived, extreme case, the difference only one third of a stop in either case. Therefore, I think you're pretty safe with using Auto WB in RAW. (Which, by the way, happened to give the same exposure as Daylight in my tests.)