Akbulut's conjecture on winding-one satellite operators

Let $*$ be either $\operatorname{ex}$ or $\operatorname{top}$, denoting the relevant concordance category, and let $P:\mathcal{C}_*\to\mathcal{C}_*$ be the map induced by a winding-number-one satellite operator $P$. Akbulut's conjecture. There exists a winding-number-one satellite operator $P$ such that $P$ is not surjective. Equivalently, by the proposition stated in the source, there exists a winding-number-one satellite operator whose induced map does not contain the unknot in its image. The conjecture concerns the surjectivity of winding-number-one satellite operators on smooth or topological concordance; the source presents it as open.