Okarun's father has it...
If the power attributed to okarun is the pinocchio, then the elements you can 100% expect to appear for sure are lies, but there is another omnipresent part of the pinocchio mythos, geppetto his father, the one who creates him from a "magical thing".
Now we have always wondered what is going on in okarun's family and house, that allows him to freely go wherever for however long, and as far as i remember never even mention needing permission, or getting into trouble or whatever you would expect from a teenager in his position, IIRC we did see his father, but only briefly and without much explanation of anything, we also only have seem his father.
So now i realize, maybe the reason they cannot extract a power from okarun, yet they refuse to believe that he truly has no power, is that the methods they use to determine where such powers are, actually points to okarun, but not because he holds the pinocchio power, but because he IS PINOCCHIO, and his father is the one who holds the power, the one who creates pinocchio, geppetto himself, thus okarun is basically a manifestation of the power, which his father controls, he is the pinocchio turned real boy, mayve okarun's father is well aware of everything going on and being either machiavelian or excentric, lets okarun deal with it on his own, so he can continue being a "real boy".