The Girth Conjecture for Heegaard splittings

Let $M$ be a closed orientable $3$-manifold and let $H\cup H^{\prime}$ be a Heegaard splitting of $M$. The algebraic girth is defined using the minimum girth of a nontrivial relator in the Heegaard splitting, while the geometric girth is the minimum girth in $H$ of an essential simple closed curve in $\partial H$ that bounds a disc in $H^{\prime}$. Girth Conjecture. If $H\cup H^{\prime}$ is not a genus two splitting of a genus two spherical manifold, then the algebraic and geometric girths of $H\cup H^{\prime}$ are equal. The algebraic girth is already known to be at most the geometric girth; the conjecture asserts the reverse inequality apart from the stated exceptional case.