The bubble-separation conjecture for two intersecting spheres

Let two spheres in $\mathbb{R}^3$ be equipped with oriented normal bundles and intersect along a circle in the space $\tilde Y$, with the canonical orientation induced on the intersection. Bubble-separation conjecture. One cannot separate two spheres intersecting along a circle while keeping their intersection a curve in $\tilde Y$. This is the basic obstruction in the proposed calculus of bubbles; the supplied text gives no proof or resolution, so its status remains open.