Phantom maps and injectivity of forgetful maps

In this note, we attack a question about the injectivity of the forgetful map posed ten years ago by Tsukiyama. We show that we can insert the forgetful map in an exact sequence and that the problem can be reduced to the computation of the sequence which turns out unexpectedly to be related to the phantom map problem and the famous Halperin conjecture in rational homotopy theory.