Primary decomposition conjecture for knot concordance

Two knots $K$ and $J$ are concordant if there is a proper locally flat embedded annulus in $S^3\times[0,1]$ with boundary $K\times\{1\}\sqcup -J\times\{0\}$; a knot is slice if it is concordant to the unknot. Their concordance classes form the knot concordance group, and each knot has an Alexander polynomial. Two polynomials are coprime if they have no nonconstant common factor.

Primary decomposition conjecture. Let $K$ and $J$ be nonslice knots. If $K$ and $J$ have coprime Alexander polynomials, then $K$ and $J$ are not concordant.

Equivalently, if the connected sum of two knots with coprime Alexander polynomials is slice, then both knots are slice. The conjecture would provide a primary decomposition of the knot concordance group analogous to the primary decomposition of Seifert forms in the algebraic concordance group; the paper presents positive evidence but does not establish the general claim.