component of $M\setminus D$ for some compact $D\subset M$.

\[def:asymcone\] Let $V$ be an end of $(M, g)$. We say that $(M, g)$ is *asymptotic to $(E_0,g_c)$ along $V$* if, for some $r > 0$, there is a diffeomorphism $\Phi:{E}_r\to V$ such that ${\lambda}^{-2}\rho_{\lambda}^*\Phi^\ast g\to g_c$ in $C^2_{\emph{loc}}(E_0, g_c)$ as ${\lambda}\to \infty$.

Using the correspondence between shrinkers and self-similar solutions, it is possible to show that any shrinker whose curvature decays quadratically on an end will be asymptotic to some regular cone on that end in the above sense and that the convergence is actually locally smooth (see, e.g., [@KotschwarWangConical] and cf. [@ChowLu]). Munteanu and Wang [@MunteanuWangConical] have shown that any complete shrinker whose Ricci curvature tends to zero at infinity must have quadratic curvature decay and must therefore be asymptotically conical along each of its ends. In dimension four, they have proven that the same is true assuming only that the scalar curvature tends to zero [@MunteanuWang4D].

The uniqueness theorem proven in [@KotschwarWangConical] asserts that if two shrinking gradient Ricci solitons are asymptotic to the same regular cone along some end of each, then the solitons are themselves isometric on some neighborhood of infinity of these ends. One consequence 