A bridge version of Gordon's conjecture

Let a bridge position of a knot or link be a bridge splitting of the pair, and call it unperturbed if it is not obtained by perturbing a lower-index bridge position. Given two unperturbed bridge positions, form their connected sum.

Bridge version of Gordon's conjecture. The connected sum of two unperturbed bridge positions is unperturbed.

The conjecture would give a direct way to establish unperturbedness for connected sums, without using a $2$-fold branched covering. It is posed in the source as a conjecture, with no resolution supplied there.