Heegaard disc-subset distance conjecture for triangulation complexity

Let $M$ be a closed orientable manifold with a Heegaard surface $S$ that divides $M$ into handlebodies $V$ and $W$. Let $\mathrm{Sp}(S)$ be the spine graph of $S$, and let $\mathcal{D}_V$ and $\mathcal{D}_W$ be the disc subsets associated to $V$ and $W$. The Heegaard disc-subset distance conjecture. The complexity $\Delta(M)$ is bounded above and below by linear functions of the distance in $\mathrm{Sp}(S)$ between $\mathcal{D}_V$ and $\mathcal{D}_W$. The conjecture proposes a complexity estimate for a manifold presented by a Heegaard splitting; the source does not state any resolution.