Pitts–Rubinstein conjecture on the Morse index of min-max limits

Let $M^3$ be a Riemannian $3$-manifold, and let a min-max limit arising from a Heegaard splitting have components $\Gamma_i$. Call a component unstable if it is unstable as a minimal surface, and let its Morse index and nullity be the dimensions of its negative and zero eigenspaces, respectively. Pitts–Rubinstein index conjecture. At most one component $\Gamma_i$ is unstable. If there is an unstable component, it has index $1$. If all components are stable, at least one component must have nullity. The unstable component, if it exists, cannot have multiplicity greater than $1$ if it is orientable, or greater than $2$ if it is nonorientable. The conjecture gives expected index, nullity, and multiplicity constraints for min-max limits from Heegaard splittings; the supplied text gives no resolution status.