Faber-Pandharipande conjecture on the prime valuations of intersection-number denominators

Let $\mathcal D_g$ be the least common multiple of the denominators of the genus-$g$ intersection numbers, and let $\operatorname{ord}(p,n)$ denote the maximum integer such that $p^{\operatorname{ord}(p,n)}\mid n$. For a prime number $p$ and $g\geq 2$, the conjecture asserts:

Faber–Pandharipande conjecture.

$$\operatorname{ord}(2,\mathcal D_g)=3g+\operatorname{ord}(2,g!),$$ $$\operatorname{ord}(3,\mathcal D_g)=g+\operatorname{ord}(3,g!),$$

and

$$\operatorname{ord}(p,\mathcal D_g)=\left\lfloor\frac{2g}{p-1}\right\rfloor\quad\text{for }p\geq 5,$$

where $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. These formulas give the conjectural exact values of the common denominators of tautological intersection numbers on the moduli space of stable curves; the source reports computations only through genus $20$, so the general assertion remains open.