6 problems

Compactification conjecture. Every $m$-dimensional subset of $\mathbb{R}^n$ has an $m$-dimensional compactification that can be embedded into $\mathbb{R}^n$.

Benítez–Sarantopoulos–Tonge conjecture. One has

3-dimensional Euclidean extension conjecture.

Plane extension conjecture.

Let $\mathcal H^u(\mathbb R^n)_b$ denote the group of bounded uniform homeomorphisms of Euclidean space $\mathbb R^n$, equipped with the sup-metric, and let $\ell_\infty$ be the Ba…

Unique-arc conjecture. The lion has a winning strategy in the lion and man game on $X$ from all starting positions if and only if any two points of $X$ are joined by a unique arc.