Existence of a composite n with n+1 dividing σ⁺(n)

Ascertain whether there exists any positive composite integer n such that n+1 divides σ⁺(n), where σ⁺(n)=∏_{p\mid n}(σ(p^{v_p(n)})+1), σ(·) is the sum-of-divisors function, and v_p(n) is the highest power of the prime p dividing n.

Background

The authors prove several negative cases for σ⁺(n), including that n+1 does not divide σ⁺(n) when n=pq or n=p^2q for primes p,q, and they establish bounds relating σ⁺(n)/(n+1) to the abundancy index. A computational search up to 10^5 revealed no composite n with n+1 dividing σ⁺(n), prompting this explicit existence question.