Dye does work but I've also seen the dyes create funny colors too. Red, rose, orangish and pink even. This has happened to me. I can remember years ago one maker saying that it had gone kinda blueish green on him. PP was what Sambar was treated with before it left India. PP is also a disinfectant not just a dye and that's what its used for more than the coloration.
To the temp question. I used PP yesterday to dye some extra large elk slabs I needed for a couple special projects. Temps were in the upper 40's and it worked exactly as it should. One soak for 45 mins gave a deep rich dark brown color. Temp doesn't seem to be an issue. These pics are from a short time ago not yesterday.
After a PP bath:
Couple of shots of clear coat:
I use quite a bit of elk, all of which I process from sheds.