Long sim sign in commutative diagram with tikz-cd

Question

In a tikzcd commutative diagram I've tried to accomplish a long sim sign (vertically, along with an arrow) but I can say that I'm dissatisfied, by far, with the result (which is given in picture below together with a M.W.E.).

The desired result is: the beginning of the long curve S, to the right of the diagram, to be at the beginning of the left vertical dashed line, and the end of this curve to be at the end of the right vertical dashed line. Also, both vertical dashed lines should be tangent to the curve (as they appear to be).

My gratitude for any good idea!

\documentclass{article}
\usepackage{tikz-cd}

\begin{document}

\[\begin{tikzcd}[row sep=1.5cm,column sep=1.5cm]
A \arrow [r] \arrow[-,d, controls={+(0.15,-1.0) and +(-0.15,1.0)},xshift=0.1cm]\arrow[d,xshift=-0.1cm]& B\arrow[-,d, controls={+(0.15,-1.0)  and +(-0.15,1.0)},xshift=0.1cm]\arrow[d,xshift=-0.1cm]\arrow[-,d,dashed,xshift=0.0cm]\arrow[-,d,dashed,xshift=0.2cm]\\
C\arrow[r]& D 
\end{tikzcd}\]

\end{document}

Answer 1

I think I found a satisfactory solution.

\documentclass{article}
\usepackage{tikz-cd}


\tikzset{
labl1/.style={anchor=north, rotate=90, inner sep=1.2mm}
}

\begin{document}

\[\begin{tikzcd}[row sep=1.25cm,column sep=1.25cm]
A \arrow [r] \arrow[d, "{\resizebox{1.4cm}{0.1cm}{$\sim$}}" labl1]& B\arrow[d, "{\resizebox{1.4cm}{0.1cm}{$\sim$}}" labl1]\\
C\arrow[r]& D 
\end{tikzcd}\]

\end{document}