A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

We prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with one of the standard cylindrical metrics on $S^k\times {\text{RR}}^{n-k}$ for $k\ge 2$ along some end must be isometric to the cylinder on that end. When the underlying manifold is complete, it must be globally isometric either to the cylinder or (when $k=n-1$) to its ${\text{ZZ}}_2$-quotient.