Poisson multinomial decomposition conjecture for floors of strongly Rayleigh variables

Let $X$ be a strongly Rayleigh random variable, and let $j,k\geq 1$. Write $\left\lfloor jX/k\right\rfloor$ for the integer-valued floor of $jX/k$. Poisson multinomial decomposition conjecture. The random variable $\left\lfloor jX/k\right\rfloor$ is a sum of independent random variables with values in $\{0,1,\ldots,j\}$. Equivalently, its probability generating function can be factorized into polynomials with positive coefficients of degrees no greater than $j$.

The claim generalizes the preceding established case $j=2$, where the floor is shown to have a Poisson multinomial representation with summands taking values in $\{0,1,2\}$. The source gives no resolution beyond posing the conjecture.