It's a hidden label switching problem.

If Laura exists, she'll ask P(blue door | laura exists). Laura=I

If Tom exists, he'll ask P(blue door | Tom exists). Tom=I

If orthonormal exists, s/he will ask P(blue door | orthonormal exists). orthonormal=I

and so on. Notice how the question we ask depends on the result of the experiment? See how the label switches?

What do Tom, Laura and orthonormal have in common? They are all conscious observers.

So, if orthonormal wakes up in a room, what orthonormal knows is that at least one conscious observer exists. P(blue room | at least one conscious observer exists)=0.5