Linear-values conjecture for σ⁺: σ⁺(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 σ⁺(n)=2p−q, where σ⁺(n)=∏_{p'\mid n}(σ(p'^{v_{p'}(n)})+1).

Background

After demonstrating structural properties of the sequence {σ⁺(n)}—including infinitely many equal-value pairs and infinitely many 3-term arithmetic progressions—the authors conjecture a strong linear-value representation property: for each fixed prime q≥3, σ⁺(n) hits the values 2p−q for infinitely many primes p, and infinitely many n.