Infinitely many primes p with φ⁺(p) < φ⁺(p+1)

Ascertain whether there exist infinitely many primes p such that φ⁺(p) < φ⁺(p+1), where φ⁺(n)=∏_{q\mid n}(φ(q^{v_q(n)})+1).

Background

Motivated by the classical open inequality for Euler’s totient function at consecutive integers, the authors prove that φ⁺(q) < φ⁺(q+1) holds for an infinite family of prime q of the form q=2p−1 (with p in OEIS A005382), and then ask whether such inequalities occur for infinitely many prime p in general.