Pitts–Rubinstein isotopy conjecture for min-max sequences

Let $M$ be a Riemannian $3$-manifold, let $H$ be a Heegaard splitting, and let a min-max sequence converge to a limiting minimal surface, possibly with multiplicity. A disk surgery is surgery on the sequence along an embedded disk, and a component may be discarded after surgery. Pitts–Rubinstein conjecture. After finitely many disk surgeries on the min-max sequence and discarding some connected components, every connected component of the min-max sequence is isotopic to a component of the limiting minimal surface or to a double cover of a component of the limiting minimal surface. This describes the possible degeneration of min-max sequences arising from Heegaard splittings; the paper proves the conjecture.