Infinitely many composites n with φ⁺(n) dividing n−1

Determine whether there exist infinitely many positive composite integers n such that φ⁺(n) divides n−1, where φ⁺(n)=∏_{p\mid n}(φ(p^{v_p(n)})+1).

Background

Building on their structural results and computations that found only n=4 up to 10^5, the authors broaden the inquiry to whether the divisibility φ⁺(n) | (n−1) occurs infinitely often among composite integers.