On the Gray index conjecture for phantom maps

We study the Gray index of phantom maps, which is a numerical invariant of phantom maps. It is conjectured that the only phantom map with infinite Gray index between finite-type spaces is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of phantom maps are restricted to simply connected finite complexes. As an application of the counter example we show that $\SNT^{\infty}(X)$ can be non-trivial for some space $X$ of finite type.