Conjectural integrality, vanishing and multinomial distribution of small-phase-space r-spin numbers

On the small phase space, write $\langle\tau_{m_1}\cdots\tau_{m_s}\rangle$ for the corresponding $r$-spin correlation number. The conjectured properties concern integers $m_1,\ldots,m_s$ and the number of insertions $s$. Conjectural properties. The following hold: (i) integrality,

$$\frac{r^{s-3}}{(s-3)!}\langle\tau_{m_1}\cdots\tau_{m_s}\rangle\in\mathbb Z;$$

(ii) vanishing, if $m_i<s-3$ for some $1\leq i\leq s$, then

$$\langle\tau_{m_1}\cdots\tau_{m_s}\rangle=0;$$

and (iii) multinomial distribution, if $m_1>m_2$, then

$$\langle\tau_{m_1-1}\tau_{m_2+1}\tau_{m_3}\cdots\tau_{m_s}\rangle\geq\langle\tau_{m_1}\tau_{m_2}\cdots\tau_{m_s}\rangle.$$

These properties are proposed as conjectural features of $r$-spin numbers on the small phase space. The supplied text does not state whether they have been proved or disproved.