Linear-values conjecture for φ⁺: φ⁺(n) = 2p − q for infinitely many pairs

Prove that for each prime q ≥ 3 there exist infinitely many pairs (p,n), with p prime, such that φ⁺(n)=2p−q, where φ⁺(n)=∏_{p'\mid n}(φ(p'^{v_{p'}(n)})+1).

Background

Echoing Conjecture 1 for σ⁺, the authors conclude with an analogous linear-value conjecture for φ⁺, suggesting that φ⁺(n) attains values of the form 2p−q for infinitely many primes p for each fixed prime q≥3.