Existence of a composite n ≠ 4 with φ⁺(n) dividing n−1

Determine whether there exists any positive composite integer n other than n=4 such that φ⁺(n) divides n−1, where φ⁺(n)=∏_{p\mid n}(φ(p^{v_p(n)})+1) and φ(·) is Euler’s totient function.

Background

The authors prove that if n=p^r with φ⁺(n) | (n−1), then necessarily n=4 (p=2,r=2). A computational check up to 10^5 found only n=4 satisfying φ⁺(n) | (n−1), motivating the explicit question whether any other composite n exists.