The satellite-operator dichotomy conjecture

For a pattern $P\subset S^1\times D^2$ and a knot $K\subset S^3$, let $P(K)$ denote the satellite knot obtained by placing $P$ in a tubular neighborhood of $K$ along the zero framing. The induced map on the knot concordance group is a satellite operator

$$P\colon \mathcal{C}\longrightarrow \mathcal{C}.$$

Satellite-operator dichotomy conjecture. Every satellite operator $P\colon \mathcal{C}\longrightarrow \mathcal{C}$ is constant or has image generating an infinite-rank subgroup of $\mathcal{C}$. This conjecture seeks to explain the observed dichotomy between satellite operators with large images and constant operators; its status is not resolved in the source.