Two-ended odd-genus self-shrinker conjecture

For sufficiently large $n\in\mathbb{N}$, consider a complete, embedded, noncompact self-shrinker $\Phi_g\subset\mathbb{R}^3$ for mean curvature flow, where $g=2n-1$ denotes its genus and the surface is invariant under the dihedral group $\mathbb{D}_n$. Two-ended odd-genus self-shrinker conjecture. For each sufficiently large $n\in\mathbb{N}$, there exists such a $\Phi_g$ with two asymptotically conical ends, and these surfaces converge locally to the union of the self-shrinking cylinder and the sphere as $n\to\infty$. The conjecture is motivated by numerical examples with odd genus and two asymptotically conical ends; it proposes a desingularisation of a self-shrinking cylinder and sphere, and remains open.