The sine formula conjecture for admissible-cover Hodge integrals

Let $I_d(g)$ be the Hodge integral over the moduli space of connected genus $g$, degree $d$ admissible covers of $\mathbb{P}^1$ with one fully ramified point and all other ramification simple, and define

$$\mathcal{I}_d(x):=\sum_{g=0}^\infty \frac{I_d(g)}{2g+d-1!}x^{2g+d-1}.$$

Sine formula conjecture. For all $d\geq 1$,

$$\mathcal{I}_d(x)=(-1)^{d-1}\frac{1}{d}\frac{\bigl(2\sin\bigl(\frac{x}{2}\bigr)\bigr)^d}{2\sin\bigl(\frac{dx}{2}\bigr)}.$$

These integrals arise as natural intersection numbers on moduli spaces of admissible covers and are related to the local Gromov--Witten theory of curves. The statement organizes the integrals into a closed generating-function formula; the supplied source gives no evidence in the provided text that the conjecture has been resolved.