Representation via φ(k(m^2 − t))/φ(ln^2) for nonsquare t

Determine whether, for each nonsquare integer t, there exist positive integers k and l such that every positive rational number can be written as φ(k(m^2 − t))/φ(ln^2) for some m, n ∈ ℕ.

Background

Theorem 2 of the paper shows that for any positive integers k and l, every positive rational number can be written as φ(k(m^2 − 1))/φ(ln^2). Together with the classical case t = 0 (i.e., φ(m^2)/φ(n^2)), these results demonstrate that the form φ(k(m^2 − t))/φ(ln^2) yields universal representations for t ∈ {0, 1}.

Motivated by this, the authors pose the open problem of whether similar universality holds for other values of t, specifically for all nonsquare integers t. They further illustrate the challenge with the special case t = −1 and ask whether all powers 2^w admit such a representation.