Celoria–Friedl–Nagel–Orson–Powell almost-concordance conjecture

Let $M$ be a closed orientable $3$-manifold with $M\neq S^3$, and let $x\in[S^1,M]$ be a free homotopy class. A dual $2$-sphere for $x$ is a $2$-sphere in $M$ representing the relevant duality obstruction; assume that $x$ does not admit one. Celoria–Friedl–Nagel–Orson–Powell conjecture. The class $x$ contains infinitely many concordance classes of knots modulo local knotting.

This predicts that almost-concordance, or concordance up to local knotting, is highly nontrivial in general $3$-manifolds. The conjecture remains open in many cases.