The asymptotic conjecture for coprime permutations

Let $C(n)$ denote the number of permutations $csigma$ of $[n]$ such that $cgcd(j,csigma(j))=1$ for every $jcin[n]$. After defining constants $c_k$ for the corresponding restrictions modulo $k!$ and setting $c_0=clim_{kctocinfty}c_k$, the asymptotic conjecture for coprime permutations.

$$C(n)=\frac{n!}{(c_0+o(1))^n}\quad\text{as }n\to\infty.$$

The constants $c_k$ arise from optimizing the limiting proportions of permitted transitions between classes determined by common divisors with the product of the primes up to $k$. The claim would identify the exponential-order asymptotic for coprime permutations, but no resolution is supplied here.