  For each positive integer $g$ we use variational methods to construct a genus
$g$ self-shrinker $\Sigma_g$ in $\mathbb{R}^3$ with entropy less than $2$ and
prismatic symmetry group $\mathbb{D}_{g+1}\times\mathbb{Z}_2$. For $g$
sufficiently large, the self-shrinker $\Sigma_g$ has two graphical
asymptotically conical ends and the sequence $\Sigma_g$ converges on compact
subsets to a plane with multiplicity two as $g\to\infty$.
Angenent-Chopp-Ilmanen conjectured the existence of such self-shrinkers in 1995
based on numerical experiments. Using these surfaces as initial conditions for
large $g$, we obtain examples of mean curvature flows in $\mathbb{R}^3$ with
smooth initial non-compact data that evolve non-uniquely after their first
singular time.
