Moments of the shifted prime divisor function

Let ω^*(n) = \{d|n: d=p-1, \mbox{pis a prime}\}. We show that, for each integer $k\ge 2$, $$\sum _{n\le x}{\omega }^{\ast }(n{)}^k\asymp x(\log x{)}^{2^k-k-1},$$ where the implied constant may depend on $k$ only. This confirms a recent conjecture of Fan and Pomerance. Our proof uses a combinatorial identity for the least common multiple, viewed as a multiplicative analogue of the inclusion-exclusion principle, along with analytic tools from number theory.