Rigidity conjecture for round cylinders in Ricci shrinkers

Let $n>0$ and $N>0$. A Ricci shrinker is a complete gradient shrinking Ricci soliton $(M^n,g,f)$. Let $d_{PGH}$ denote pointed Gromov–Hausdorff distance, and let $\Gamma$ act freely on $S^{n-1}$ with $|\Gamma|\leq N$. The notation $(S^{n-1}/\Gamma)\times\mathbb{R}$ denotes the corresponding round cylinder with base point $p_c$ and metric $g_c$.

Round-cylinder rigidity conjecture. There exists a small constant $\epsilon=\epsilon(n,N)>0$ such that, if

$$d_{PGH}\left\{(M^n,p,g),\left((S^{n-1}/\Gamma)\times\mathbb{R},p_c,g_c\right)\right\}<\epsilon,$$

then $(M,g)$ is isometric to $\left((S^{n-1}/\Gamma)\times\mathbb{R},p_c,g_c\right)$.

This conjecture, originally proposed in the cited work of L. Wang and W. Wang, asserts rigidity of Ricci shrinkers sufficiently close to a round cylinder. The paper states that it remains open; a weaker version under the assumption that $f$ has no critical point outside a compact set is proved.