Hedden's conjecture on homomorphisms induced by satellite patterns

A pattern $P\subset S^1\times D^2$ induces a map $[P]\colon\mathcal{C}\to\mathcal{C}$ on the knot concordance group, given by $[K]\mapsto[P(K)]$. This map is a homomorphism when $P(K\mathbin{\#}K')$ is concordant to $P(K)\mathbin{\#}P(K')$ for all knots $K,K'$. Hedden's conjecture. If $[P]$ is a homomorphism, then it is one of the maps

$$[K]\mapsto[U],\qquad [K]\mapsto[K],\qquad [K]\mapsto[K^r].$$

These are the three evident homomorphisms induced by satelliting; the conjecture asserts that no other satellite pattern induces a homomorphism of the concordance group.