Crossing-arc separation conjecture for connected sums

Let $K\#K'$ be a connected sum of knots, let a collection of unknotting crossing arcs for $K\#K'$ be given, and let the 2-sphere specifying the connected sum be fixed. Crossing-arc separation conjecture. In any collection of unknotting crossing arcs for $K\#K'$, there is one arc that can be isotoped to be disjoint from the 2-sphere specifying the connected sum. This is presented as a stronger conjectural approach to the additivity conjecture for unknotting number. The source says that the authors found no counterexample to additivity but does not establish this stronger statement.