Asymptotic equality of Carmichael and weak Carmichael numbers

Let $C(n)$ and $W(n)$ denote the numbers of Carmichael numbers and weak Carmichael numbers in the interval $[1,n]$, respectively. Asymptotic equality conjecture.

$$C(n)\sim W(n)\qquad\text{as }n\to\infty.$$

The conjecture predicts that weak Carmichael numbers outside the Carmichael numbers are asymptotically negligible.