= Solution
An affine <geodesic> has constant squared tangent norm, but that constant depends on the scale of its parameter. Write
$$
\mathcal N=g_{AB}\dot x^A\dot x^B=\text{constant},\qquad
du^2=\eta_{ab}dx^a dx^b.
$$
Using $W=\xi$ in the line element gives
$$
\left(\frac{du}{d\lambda}\right)^2=\mathcal N-\xi^2,
\qquad
\boxed{du^2=(\mathcal N-\xi^2)d\lambda^2.}
$$
The <affine normalization in a Kaluza-Klein geodesic reduction> matters: the specific formula requested in the paper uses $\mathcal N=1$, namely $\lambda$ equal to the positive five-dimensional arclength. With that normalization,
$$
\boxed{d\tilde u^2=\frac{du^2}{1-\xi^2}.}
$$
A timelike four-dimensional projection has $du^2=c^2d\tau^2>0$, requiring $|\xi|<1$ in this normalization. At $|\xi|=1$ the projection is null and has no proper-time <four-velocity>. For an arbitrary <affine parameter> the first boxed relation, with $\mathcal N$ retained, is the general result.
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