 Solution

False. Use the same flat-point construction for both curves and make their tangent fields identical for $s<0$. For $s>0$, let one curve bend in the $xz$-plane and let the other bend in the plane spanned by $(1,0,0)$ and $(0,\cos \theta ,\sin \theta )$, where this second plane is neither the $xz$-plane nor its reflection fixing the $xy$-plane. The two curves have the same curvature everywhere and zero torsion away from $s_0=0$.

Any rigid motion relating them parameterwise would be fixed by their common nonstraight arc for $s<0$; its orthogonal part can only fix the $xy$-plane pointwise and possibly reflect its normal. It therefore cannot carry the chosen positive-side plane to the other one. The curves need not be related by a rigid motion.