The primitive curve conjecture for Heegaard splittings

Let $M$ be a closed orientable $3$-manifold and let $H\cup H^{\prime}$ be a Heegaard splitting of $M$. A closed curve in $\partial H$ is primitive in $H$ if it does not represent a proper power in $\pi_1(H)$. Primitive curve conjecture. If $\gamma$ is a closed curve in $\partial H$ that is primitive in $H$ and contracts in $H^{\prime}$, then there is an essential simple closed curve $\gamma^{\prime}$ in $\partial H$ with girth no greater than that of $\gamma$ which contracts in $H^{\prime}$. This proposes that algebraically simple compressing curves can be replaced by essential simple closed compressing curves without increasing girth.