Macdonald strong factorization and cumulant conjecture

Let $\lambda^1,\dots,\lambda^r$ be partitions. For $I\subseteq [r]$, let $\lambda^I$ denote the combined partition used in the Macdonald polynomial $J_{\lambda^I}^{(q,t)}$, and let $\kappa^M(\lambda^1,\dots,\lambda^r)$ denote the cumulant of the Macdonald polynomials $J^{(q,t)}$. Macdonald cumulant conjecture. For any partitions $\lambda^1,\dots,\lambda^r$, as $q\to 1$, both the strong factorization property

$$\prod_{I\subset [r]}\left(J_{\lambda^I}^{(q,t)}\right)^{(-1)^{|I|}}=1+O\left((q-1)^{r-1}\right)$$

and the cumulant estimate

$$\kappa^M(\lambda^1,\dots,\lambda^r)=O\left((q-1)^{r-1}\right)$$

hold. The paper reports that this analogue of the Jack-polynomial factorization property was suggested by computer simulations but could not be proved there; no resolution is supplied.