Existence of full finite-entropy ancient mean curvature flows

Let $N$ and $n$ be positive integers with $N\geq n$. An ancient solution to the mean curvature flow in $\mathbb{R}^N$ is full if it does not lie in any $(N-1)$-dimensional Euclidean subspace.

Existence conjecture. There exists an $n$-dimensional, connected, full ancient solution to the mean curvature flow in $\mathbb{R}^N$ with finite entropy.

The preceding theorem establishes this existence when $n\geq2$, while the conjecture extends the assertion to the remaining case allowed by $N\geq n$. The construction is notable for producing explicit ancient flows in Euclidean space with finite entropy.