I actually watched a video on YouTube where a guy tested a bunch of different car fog lights and in his testing, amber did in fact lose less light, but it was a small percentage difference and the white lights still got more light down range because they started off with more light. Cutting light output to lose less light output kind of defeats the purpose.
Aim is definitely your greatest ally with fog, but there is validity to warmer lights being better. You're literally having a light shining into your eyes and whether it's reflective particles or somebody else's headlights, warmer lights feel less harsh.
EDIT: Found the video