How stable is Levinson-Durbin method for solution of systems of linear equations ?

I mean if condition number of matrix is $k$, does intermidiate steps involve matrixes with higher condition number ? For example QR is easy to see preserve condition number so it is stable, but Cholesky for example increase condition number from $k$ to $k^2$, so Cholesky is less stable.

The question is related with possible fix-point implementation of this method and we need to understand how many bits should be given, for matrices of small size.