Intersection of prime imbalances with their Möbius image

Determine whether the intersection of the set of prime imbalances I := { |p − q| / (p + q) : p, q ∈ ℙ, p > q } with its image under the Möbius transformation μ(x) = (1 − x) / (1 + x) equals {2/5, 3/7}; equivalently, ascertain whether any rational imbalance δ = |p − q| / (p + q) for primes p > q other than 2/5 and 3/7 satisfies μ(δ) = |r − s| / (r + s) for some primes r > s.

Background

The paper defines the imbalance between two primes p > q as δ(p, q) = |p − q| / (p + q), and considers the set of all such rational values I. It introduces the Möbius transform μ(x) = (1 − x) / (1 + x) acting on these imbalances and studies the overlap I ∩ μ(I).

Within a proposition analyzing I ∩ μ(I), the authors present empirical evidence up to N = 200 showing that only the values 2/5 and 3/7 mutually map into I under μ, and they provide explicit examples for these two cases. They then explicitly conjecture that no other overlaps occur, motivating a precise determination of whether I ∩ μ(I) is exactly {2/5, 3/7}.