Multinomial monotonicity conjecture for intersection numbers of stable curves

For nonnegative integers $d_1,\ldots,d_n$ satisfying

$$\sum_{i=1}^{n}d_i=3g-3+n,$$

and with $d_1<d_2$, let $\langle\tau_{d_1}\cdots\tau_{d_n}\rangle_g$ denote the genus-$g$ intersection number of descendant classes on the moduli space of stable curves.

Multinomial monotonicity conjecture.

$$\langle\tau_{d_1}\tau_{d_2}\cdots\tau_{d_n}\rangle_g\leq\langle\tau_{d_1+1}\tau_{d_2-1}\cdots\tau_{d_n}\rangle_g.$$

The inequality is motivated by the genus-zero multinomial formula and is proved in the source for genus $1$; it is conjectured here for arbitrary genus, with the stated dimension constraint.