Removing the balance condition in Peck’s conjecture

Determine whether Theorem 1.1 remains valid without the condition that the auxiliary functions satisfy the balance condition (1.2), namely that the square of the largest parameter is at most the smallest parameter.

Background

Theorem 1.1 establishes Peck’s conjecture for simultaneous Diophantine approximation under the additional condition (1.2) on the parameters governing the approximation functions. The authors explain that this condition yields, for example, logarithmic factors of the form $(\log Q)^{\nu_i}$ when the exponents are sufficiently balanced.

The unresolved issue is whether the same conclusion holds for arbitrary admissible functions satisfying the product condition, without imposing the balance restriction between their sizes.