Hironaka's asymptotic dilatation conjecture for closed surfaces

For each genus $g$, let $\delta_{g,0}$ be the minimum dilatation of pseudo-Anosov maps on the closed surface of genus $g$. Let $\mu=\frac{1+\sqrt{5}}{2}\approx1.618$ be the golden ratio. Hironaka's conjecture. The minimum dilatations grow so that

$$\lim_{g\to\infty}\delta_{g,0}^g=\mu^2\approx2.618.$$

This is a well-known asymptotic conjecture about minimum dilatations on closed surfaces. The source gives no resolution and places it among the open minimum-dilatation problems for surface homeomorphisms.