Celoria–Levine–Friedl–Nagel–Orson–Powell concordance conjecture

Let $M\neq S^3$ be a closed orientable $3$-manifold, let $[S^1,M]$ denote the set of free homotopy classes of loops in $M$, and let $\mathcal{C}^M_x$ be the set of concordance classes of knots in $M$ representing $x\in[S^1,M]$. An embedded dual 2-sphere for $x$ is an embedded $2$-sphere in $M$ meeting a representative of $x$ transversely in one point. Concordance conjecture. The concordance set $\mathcal{C}^M_x$ contains infinitely many elements if and only if the class $x$ does not admit an embedded dual 2-sphere. The Concordance Lightbulb Theorem establishes the complementary case: when $x$ admits an embedded dual $2$-sphere, all knots representing $x$ are concordant. The conjecture asserts that this is the exceptional case, while infinitude in the remaining cases is open in general.