Ribbon knots and iterated cables of fibered knots

We define a knot to be ${\gamma }_0$-sharp if its Seifert genus is detected by the concordance invariant ${\gamma }_0$, which arises from the immersed curve formalism in bordered Heegaard Floer homology. We show that a connected sum of ${\gamma }_0$-sharp fibered knots is ribbon exactly when it is of the form $K\mathrm{\#}-K$. Consequently, either iterated cables of tight fibered knots are linearly independent in the smooth concordance group, or the slice–ribbon conjecture fails.